Home » Mathematics » Pre Calculus » Idea of a Limit

Vincent Selhorst-Jones

Vincent Selhorst-Jones

Idea of a Limit

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Table of Contents

Section 1: Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
Section 2: Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
Section 3: Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
Section 5: Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
Section 6: Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Section 7: Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
Section 8: Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
Section 10: Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
Section 11: Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
Section 13: Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
Section 14: Conic Sections
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Section 15: Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
Section 16: Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
Section 17: Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Lecture Comments (10)

1 answer

Last reply by: Professor Selhorst-Jones
Tue Nov 19, 2013 10:55 AM

Post by Joel Fredin on November 13, 2013

Your teaching methods are insanely good. You are the god of teaching, seriously.

1 answer

Last reply by: Professor Selhorst-Jones
Fri Aug 16, 2013 4:12 PM

Post by Jorge Sardinas on August 15, 2013

for example 3 isn't 0/0 0 instead of 1 if not can you please say why Mr Jones?

3 answers

Last reply by: Professor Selhorst-Jones
Tue Aug 13, 2013 10:16 AM

Post by Taylor Wright on August 12, 2013

Please make a full series on Calculus! You are an amazing teacher!

1 answer

Last reply by: Professor Selhorst-Jones
Sun Apr 14, 2013 6:30 PM

Post by Orsolya Krispán on April 14, 2013

I can't stop loving your teachings methods! yeah, it really did freak me out at the 4th example. :D thank you guys for everything, you all really did help me a lot, and please cross your fingers for me to have a successful final exam and get into my dream university. Thanks a lot!

Idea of a Limit

  • This lesson marks our entry into an entirely new section of mathematics: calculus. From here on, this course will preview some of the topics you will be exposed to in a calculus class.
  • We can conceive of a limit as the vertical location a function is "headed towards" as it gets closer and closer to some horizontal location. Equivalently, a limit is what the function output is "going to be" as we approach some input.
  • For notation, we use

    lim
     
    		f(x)
    to say, "the limit of f(x) as x approaches c."
  • Another way to conceive of a limit is to imagine "covering up" the location we're approaching with a thin strip. With that location covered, we ask, "Where does it look like this function is going (now that we can't see where it actually winds up)?"
  • Definition of a limit: If f(x) becomes arbitrarily close to some number L as x approaches some number c (but is not equal to c), then the limit of f(x) as x approaches c is L. Symbolically,

    lim
     
    		f(x) = L.
  • In the above definition, there are two important things to note:
    • We are looking at f(x) as x→ c, but we are not concerned with f(x) at x=c.
    • When we consider x approaching c (x→ c), we are considering x approaching from all directions, not just one side. To have a limit, f(x) must go to the same value from both sides.
  • It should be pointed out that the above is not technically the formal definition of a limit. This will certainly be enough for now, but if you're curious to know about it, check out the next lesson, Formal Definition of a Limit. [But you won't need to understand that for a couple years (if ever), probably.]
  • Limits do not always exist. If the two sides don't settle down towards the same location, there will be no limit.
  • There are three main ways to find limits:
    • Graphs: Look at a graph of the function, and figure out if it makes sense to have a limit at the location. If so, find out what value the graph indicates. [This method is not very precise, but it gives a pretty good idea.]
    • Tables: Make a table of values where the input values get really close to the location we approach in the limit. If the output values settle down towards a single value the closer we get to the location, there is a limit. [This method is more precise and can give very good approximations, but is still not perfectly accurate.]
    • Precise Methods: Later on, we will see algebraic methods to find precise values for limits. The lesson Finding Limits will go over this in detail. For right now, though, we'll just stick to the methods of graphing and making tables.

Idea of a Limit

Fill in the table of values below (notice that you do not fill in the spot with the `?' symbol). Use your result to determine if the limit exists-if it does, estimate the limit.

lim
x→ 2 
 4x−3

x
1.9
1.99
1.999
2
2.001
2.01
2.1
f(x)
             
             
             
    ?    
             
             
             
  • We fill in the table exactly like we would fill in any table of values. We have various x-values that we use as input, then we put down the corresponding output value. [Don't be confused by the fact that the table lists `f(x)' and we haven't explicitly named a function: the problem is implying that what comes after the `lim' is the function since it's the only thing that looks like a function.]
  • Plug in each value, see what the result is. Here are the first couple values:
    x=1.9     ⇒     4(1.9) −3     =     4.6

    x=1.99     ⇒     4(1.99) −3     =     4.96
    We do the rest of the x-values in the same manner, filling in the table as we go. Once completed, we have the below:
    x
    1.9
    1.99
    1.999
    2
    2.001
    2.01
    2.1
    f(x)
        4.6    
        4.96    
        4.996    
        ?    
        5.004    
        5.04    
        5.4    
  • Now that we have the table filled in, we can work on the limit. [If you didn't watch the video, make sure to do so. The idea of a limit is quite special and it's extremely important to understand the concept.] Remember, a limit is a way to ask what value the function approaches as x goes to a specific value. In this case, we have the limit as x→ 2, so we want to know what value the function is going to as x gets very close to 2. [Notice that we do not care what the actual value is at x=2. A limit is based on what happens near x=2, not at x=2. This is why there is a `?' symbol for x=2 on the table: because it does not affect the limit.] Look at the table to see what value the function is approaching. As x gets closer and closer to 2, it appears that f(x) → 5. The closer x is to 2, the closer f(x) is to 5. Therefore we have that the limit exists and that the limit equals 5.

x
1.9
1.99
1.999
2
2.001
2.01
2.1
f(x)
    4.6    
    4.96    
    4.996    
    ?    
    5.004    
    5.04    
    5.4    
Limit exists;    limx→ 2 4x−3   = 5
Fill in the table of values below (notice that you do not fill in the spot with the `?' symbol). Use your result to determine if the limit exists-if it does, estimate the limit.

lim
x→ −4 
 x2+3x+3

x
−4.1
−4.01
−4.001
−4
−3.999
−3.99
−3.9
f(x)
             
             
             
    ?    
             
             
             
  • We fill in the table exactly like we would fill in any table of values. We have various x-values that we use as input, then we put down the corresponding output value. [Don't be confused by the fact that the table lists `f(x)' and we haven't explicitly named a function: the problem is implying that what comes after the `lim' is the function since it's the only thing that looks like a function.]
  • Plug in each value, see what the result is. Here are the first couple values:
    x=−4.1     ⇒     (−4.1)2+3(−4.1)+3     =     7.51

    x=−4.01     ⇒     (−4.01)2+3(−4.01)+3     =     7.0501
    We do the rest of the x-values in the same manner, filling in the table as we go. Once completed, we have the below:
    x
    −4.1
    −4.01
    −4.001
    −4
    −3.999
    −3.99
    −3.9
    f(x)
        7.51   
        7.05   
        7.005   
        ?    
        6.995   
        6.95   
        6.51   
  • Now that we have the table filled in, we can work on the limit. [If you didn't watch the video, make sure to do so. The idea of a limit is quite special and it's extremely important to understand the concept.] Remember, a limit is a way to ask what value the function approaches as x goes to a specific value. In this case, we have the limit as x→ −4, so we want to know what value the function is going to as x gets very close to −4. [Notice that we do not care what the actual value is at x=−4. A limit is based on what happens near x=−4, not at x=−4. This is why there is a `?' symbol for x=−4 on the table: because it does not affect the limit.] Look at the table to see what value the function is approaching. As x gets closer and closer to −4, it appears that f(x) → 7. The closer x is to −4, the closer f(x) is to 7. Therefore we have that the limit exists and that the limit equals 7.

x
−4.1
−4.01
−4.001
−4
−3.999
−3.99
−3.9
f(x)
    7.51   
    7.05   
    7.005   
    ?    
    6.995   
    6.95   
    6.51   
Limit exists;    limx→ −4 x2+3x+3   = 7
Fill in the table of values below (notice that you do not fill in the spot with the `?' symbol). Use your result to determine if the limit exists-if it does, estimate the limit.

lim
x→ 5 
 x2−9x+20
x−5

x
4.9
4.99
4.999
5
5.001
5.01
5.1
f(x)
             
             
             
    ?    
             
             
             
  • We fill in the table exactly like we would fill in any table of values. We have various x-values that we use as input, then we put down the corresponding output value. [Don't be confused by the fact that the table lists `f(x)' and we haven't explicitly named a function: the problem is implying that what comes after the `lim' is the function since it's the only thing that looks like a function.]
  • Plug in each value, see what the result is. Here are the first couple values:
    x=4.9     ⇒     (4.9)2−9(4.9)+20
    (4.9)−5
    		    =     0.9

    x=4.99     ⇒     (4.99)2−9(4.99)+20
    (4.99)−5
    		    =     0.99
    We do the rest of the x-values in the same manner, filling in the table as we go. Once completed, we have the below:
    x
    4.9
    4.99
    4.999
    5
    5.001
    5.01
    5.1
    f(x)
        0.9    
        0.99    
        0.999    
        ?    
        1.001    
        1.01    
        1.1    
  • Now that we have the table filled in, we can work on the limit. [If you didn't watch the video, make sure to do so. The idea of a limit is quite special and it's extremely important to understand the concept.] Remember, a limit is a way to ask what value the function approaches as x goes to a specific value. In this case, we have the limit as x→ 5, so we want to know what value the function is going to as x gets very close to 5. [Notice that we do not care what the actual value is at x=5. A limit is based on what happens near x=5, not at x=5. This is why there is a `?' symbol for x=5 on the table: because it does not affect the limit. This is especially important in this case since f(5) does not exist because it would give [0/0].] Look at the table to see what value the function is approaching. As x gets closer and closer to 5, it appears that f(x) → 1. The closer x is to 5, the closer f(x) is to 1. Therefore we have that the limit exists and that the limit equals 1.

x
4.9
4.99
4.999
5
5.001
5.01
5.1
f(x)
    0.9    
    0.99    
    0.999    
    ?    
    1.001    
    1.01    
    1.1    
Limit exists;    limx→ 5 [(x2−9x+20)/(x−5)]   = 1
Fill in the table of values below (notice that you do not fill in the spot with the `?' symbol). Use your result to determine if the limit exists-if it does, estimate the limit.

lim
x→ 1 
 2
x−1

x
0.9
0.99
0.999
1
1.001
1.01
1.1
f(x)
             
             
             
    ?    
             
             
             
  • We fill in the table exactly like we would fill in any table of values. We have various x-values that we use as input, then we put down the corresponding output value. [Don't be confused by the fact that the table lists `f(x)' and we haven't explicitly named a function: the problem is implying that what comes after the `lim' is the function since it's the only thing that looks like a function.]
  • Plug in each value, see what the result is. Here are the first couple values:
    x=0.9     ⇒     2
    x−(0.9)
    		    =     −20

    x=0.99     ⇒     2
    x−(0.99)
    		    =     −200
    We do the rest of the x-values in the same manner, filling in the table as we go. Once completed, we have the below:
    x
    0.9
    0.99
    0.999
    1
    1.001
    1.01
    1.1
    f(x)
        −20    
        −200    
        −2000    
        ?    
        2000    
        200    
        20    
  • Now that we have the table filled in, we can work on the limit. [If you didn't watch the video, make sure to do so. The idea of a limit is quite special and it's extremely important to understand the concept.] Remember, a limit is a way to ask what value the function approaches as x goes to a specific value. In this case, we have the limit as x→ 1, so we want to know what value the function is going to as x gets very close to 1. [Notice that we do not care what the actual value is at x=1. A limit is based on what happens near x=1, not at x=1. This is why there is a `?' symbol for x=1 on the table: because it does not affect the limit. This is especially important in this case since f(1) does not exist because it would give [2/0].] Look at the table to see what value the function is approaching. As x gets closer and closer to 1, we see that it shoots off to −∞ on the left side (x < 1) and shoots off to ∞ on the right side (x > 1). The value of the function is not converging to a single value as x→1: instead it is flying off in opposite directions. Therefore the limit does not exist.

x
0.9
0.99
0.999
1
1.001
1.01
1.1
f(x)
    −20    
    −200    
    −2000    
    ?    
    2000    
    200    
    20    
Limit does not exist: the function does not steadily approach a single value as x→1.
Using the associated graph below, determine if the limit exists. If so, give the value of the limit.

lim
x→ 1 
 −2x2+2x
x3−x2+x−1
  • The limit of a function is the value that the function approaches as the function goes toward a certain x-value. For this problem, we're interested in figuring out if the function is approaching a single value as x→ 1.
  • Look at the graph to see if the limit exists. Looking in the "neighborhood" around x=1, we see that both the left and right portions of the graph are being pulled towards a single value. Left of x=1 and right of x=1 are both heading towards where the empty circle is. Thus, because the function is clearly headed to the same place from both sides, the limit exists. [Notice that we do not care that the empty circle indicates the function does not exist at x=1 (it would give [0/0]). The limit only cares about what the function is headed towards, not what it actually is at the x-value. Thus we only care about what the left and right sides are headed towards.]
  • The value of the limit is what the function is headed towards as x→ 1. Previously, we saw that both sides are headed towards the empty circle. Thus, we only need to find what the height of the empty circle is to find the value of the limit. Carefully looking at the graph, we see that it is at a height of −1. Thus, as x→ 1, the function approaches a height of −1, so the limit has a value of −1.
Limit exists;    limx→ 1 [(−2x2+2x)/(x3−x2+x−1)]   = −1
Using the associated graph below, determine if the limit exists. If so, give the value of the limit.

lim
x→ 3 
 2x−6
x2−4x+3
  • The limit of a function is the value that the function approaches as the function goes toward a certain x-value. For this problem, we're interested in figuring out if the function is approaching a single value as x→ 3.
  • Look at the graph to see if the limit exists. Looking in the "neighborhood" around x=3, we see that both the left and right portions of the graph are being pulled towards a single value. Left of x=3 and right of x=3 are both heading towards where the empty circle is. Thus, because the function is clearly headed to the same place from both sides, the limit exists. [Notice that we do not care that the empty circle indicates the function does not exist at x=3 (it would give [0/0]). The limit only cares about what the function is headed towards, not what it actually is at the x-value. Thus we only care about what the left and right sides are headed towards.]
  • The value of the limit is what the function is headed towards as x→ 3. Previously, we saw that both sides are headed towards the empty circle. Thus, we only need to find what the height of the empty circle is to find the value of the limit. Carefully looking at the graph, we see that it is at a height of 1. Thus, as x→ 3, the function approaches a height of 1, so the limit has a value of 1.
Limit exists;    limx→ 3 [(2x−6)/(x2−4x+3)]   = 1
Using the associated graph below, determine if the limit exists. If so, give the value of the limit.

lim
x→ 2 
 g(x),    where g(x) =⎧
⎪
⎨
⎪
⎩
2x,
x ≤ 2
− 1
2
 x + 5,
x > 2
  • The limit of a function is the value that the function approaches as the function goes toward a certain x-value. For this problem, we're interested in figuring out if the function is approaching a single value as x→ 2.
  • Look at the graph to see if the limit exists. Looking in the "neighborhood" around x=2, we see that both the left and right portions of the graph are being pulled towards a single value. Left of x=2 and right of x=2 are both heading towards the top "corner" of the graph. Thus, because the function is clearly headed to the same place from both sides, the limit exists.
  • The value of the limit is what the function is headed towards as x→ 2. Previously, we saw that both sides are headed towards the "corner" at the top. Thus, we only need to find what the height of that location is to find the value of the limit. Carefully looking at the graph, we see that it is at a height of 4. Thus, as x→ 2, the function approaches a height of 4, so the limit has a value of 4.
Limit exists;    limx→ 2  g(x)   = 4
Using the associated graph below, determine if the limit exists. If so, give the value of the limit.

lim
x→ −3 
  (x+1)3
30(x+3)
  • The limit of a function is the value that the function approaches as the function goes toward a certain x-value. For this problem, we're interested in figuring out if the function is approaching a single value as x→ −3.
  • Look at the graph to see if the limit exists. Looking in the "neighborhood" around x=−3, we see that the left and right portions of the graph are going to totally different places. For the part of the graph that is left of x=−3, it's flying off towards ∞, while the part that is right of x=−3 is flying off towards −∞. Thus, because the two sides of the function do not "agree" on a single value they are both headed towards, the limit does not exist
  • Because the limit does not exist, it has no value. We cannot assign a value to something that does not exist.
Limit does not exist.
Draw a graph of the below function, then use that to determine if the limit exists and, if so, what its value is.

lim
x → 3 
 f(x),    where f(x) =⎧
⎪
⎨
⎪
⎩
−2x+3,
x ≤ 3
(x−2)2 − 4,
x > 3
  • Begin by drawing a graph of the piecewise function g(x). [If you're unfamiliar with piecewise functions or need a refresher, check out the lesson Piecewise Functions in the section on Functions. That lesson will fully explain the idea of a piecewise function along with showing how to graph them.]
  • When graphing the piecewise function, you can create a table of values to help you see how to plot points and create the graph.

    Another alternative is to create another graph that plots each "piece" of the function without regard to the interval the "piece" is graphed on. This makes it easier to see how each "piece" is a part of the whole graph of the piecewise function. We can see this below, where −2x+3 is graphed in red and (x−2)2−4 is graphed in blue. Once you see how each "piece" is graphed, erase everything except the interval that it is used on. For example, the right side of −2x+3 will be erased so the only part remaining is for the section where x ≤ 3. We can see the completed graph of the piecewise function f(x) on the next step.
  • However you create the graph of f(x), you should get the below [The graph below has been color-coded to help you see each individual "piece" that makes it up. This is not necessary, since the graph of f(x) is both "pieces" taken together, but it does make it slightly easier to understand how f(x) creates its graph.]:
  • Now that you have a graph of f(x), we want to consider if it has a limit as x→ 3. To decide this, we need to see if the function is approaching a single value from both sides in the neighborhood around x=3. Looking at the graph, we see that both the part to the left of x=3 and to the right of x=3 are headed to the same location. Therefore the function has a limit as x→ 3. The height of the location being approached is −3, so that is the value of the limit.
Limit exists;    limx → 3  f(x)    =  −3
Below is the graph of g(x) = {
x2 −8,
x ≤ −1
x2 − 8,
x ≥ 1
. Explain why limx→ 0 g(x)  does not exist.
  • Start off by noticing that g(x) has a gap in its graph. There's nothing on the graph for the x-interval (−1, 1). Why? Because the function is not defined over this interval. Look at how the piecewise function is defined. We know what to do for x ≤ −1 and we know what to do for x ≥ 1, but it says nothing for −1 < x < 1. Thus, on that interval, the function is not defined so it does not exist for any of the values in that interval.
  • Next, consider that a limit is the value that a function approaches as the x-value approaches some specific number. For this problem, we're interested in seeing what value g(x) approaches as x→ 0
  • Finally, notice that the function is not approaching any value as x→0. While we can "see" what happens for x ≤ −1 and x ≥ 1, because the function is not defined for (−1, 1), there's simply nothing happening inside the gap. Because of this, there is no limit point to approach. The function is not defined in that "neighborhood", so there is no limit. [Notice that this is different from the function not being defined at only one point. While a limit is unaffected by what the function's value (or lack of value) is at the specific x-value being approached, it still must be defined in the vicinity around that x-value.]
The function is not defined on the x-interval of (−1, 1), so it is impossible for the function to approach any values while using that interval, and thus no limits can exist for x-values in that interval.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Idea of a Limit

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Introduction 0:05
  • Motivating Example 1:26
  • Fuzzy Notion of a Limit 3:38
    • Limit is the Vertical Location a Function is Headed Towards
    • Limit is What the Function Output is Going to Be
    • Limit Notation
  • Exploring Limits - 'Ordinary' Function 5:26
    • Test Out
    • Graphing, We See The Answer Is What We Would Expect
  • Exploring Limits - Piecewise Function 6:45
    • If We Modify the Function a Bit
  • Exploring Limits - A Visual Conception 10:08
  • Definition of a Limit 12:07
    • If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
    • We Are Not Concerned with f(x) at x=c
    • We Are Considering x Approaching From All Directions, Not Just One Side
  • Limits Do Not Always Exist 15:47
  • Finding Limits 19:49
    • Graphs
    • Tables
    • Precise Methods
  • Example 1 26:06
  • Example 2 27:39
  • Example 3 30:51
  • Example 4 33:11
  • Example 5 37:07

Transcription: Idea of a Limit

Hi--welcome back to Educator.com.0000

Today, we are going to talk about the idea of a limit.0002

This lesson marks our entry into an entirely new section of mathematics; we are entering calculus territory.0005

From here on, this course will preview some of the topics that you will be exposed to in a calculus class.0011

You might wonder what calculus is and why it matters.0016

In short, calculus is a new way to look at functions.0021

It gives us new tools to analyze function behavior and see how they relate to one another and the world around us.0025

As to why it matters: calculus is crucially important to science and engineering.0031

If you want to get anything of any real depth done in those fields, you need calculus in your tool belt.0036

Plus, I think it is just really cool; it is one of these really cool things that you can do in mathematics.0041

It is a really new, interesting idea, and we get to play around with it and do all sorts of cool stuff.0045

And it creates...from here one, we get to see a whole bunch of new stuff that lets us explain a whole bunch of phenomena in the real world--really cool stuff.0050

For the next few lessons, we will focus on limits.0057

Limits allow us to describe functions in ways that were previously impossible for us to describe.0060

We will be able to approach the infinite and the infinitesimal (that being the infinitely small) things with them.0065

And they make up the heart of calculus; they are pretty much the foundation that the rest of calculus rests upon.0071

So, it is really useful to have a good understanding of what a limit is and how it works,0077

before we can start moving into other ideas in calculus that have more direct applicability to the daily world.0080

All right, let's start with a motivating example to get things started.0087

Consider the function f(x) = x/x: what happens if we try to look at f(0)?0091

Well, if we plug 0 in, then we are going to get f(0) = 0/0; but since dividing by 0 is not defined, f(0) does not exist.0098

Therefore, we cannot evaluate the value of the function at 0; f(0) is not a thing.0110

We can't do anything with it, and that is the end of the story--that is it.0114

And up until now, that would have been the end of the story.0119

But now, when we look at f(0), well, OK, it is kind of the end of the story on what f(0) is.0122

But before we entirely refuse the idea of considering f at 0, considering how f and 0 interact, let's look at the function's graph.0131

If we look at f(x) = x/x, look at that: the function is always 1.0139

Now, right here at x = 0 on our vertical y-axis, we have this hole here.0145

That circular hole tells us that it does not actually exist there.0152

But anywhere that isn't x = 0, we end up getting a 1 out of it.0155

On the one hand, f(0) does not exist; you plug in a 0; you get 0/0; that is bad.0160

You are not allowed to divide by 0, so we say that f(0) does not exist.0166

But on the other hand, it is obvious where it is headed: look, the thing is going right in towards that 1.0169

On the one hand, sure, it doesn't exist; but it totally should be a 1.0176

So, we cannot say f(0) = 1, because f(0) does not exist, remember.0182

But we still want some way to talk about where it was headed.0188

It was clearly going to be a 1 before those pesky rules about dividing by 0 got in the way and stopped us0192

from being able to figure out what the real answer should have been--0199

what we feel like it was going towards, at least--perhaps not the real answer, because...0203

we will talk about it; you will see other things about why we can't really assign a direct value to it.0208

But we can talk about this idea of where it was headed; and we talk about that with a limit.0212

With this idea in mind, we want to think of a limit as where it was headed.0219

A limit is the vertical location that a function is headed towards--what it is going to be at as it gets closer and closer to some horizontal location.0223

In that previous example, our motivating one of x/x, we had that, as it got closer and closer to 0, it got closer and closer to being at a height of 1.0236

In fact, it was always at a height of 1, no matter where it was.0244

We end up seeing this sense of...as it gets closer to 0, it is really, really close to this value, where it is headed towards.0248

Equivalently, a limit is what the function output is going to be as we approach some input.0255

As we get close to some input, what output seems like it is going to come out of it--what should it be; what is it going to be?0263

We are going to be talking about this idea a lot, so we want some notation to describe it.0273

We use this notation right here: this part here, the "lim," says that it is the limit we are working with.0278

The x arrow c says what we are going towards; and the f(x) is the thing that actually is having the limit be applied to it; that is the function.0285

This says "the limit of f(x) as x approaches c"; so as x gets close to c, what happens to f(x).0292

You might also hear this spoken aloud as "the limit as x goes to c of f(x)"; I tend to say that a lot.0302

Or you might hear some similar variant; in any case, the idea is the same.0308

It is this question of what f(x) does as x gets very close to c.0311

x is going to c, and our question is, "What will f(x) do in response to x going to c?"0317

Let's test out our new idea on an old friend, f(x) = x2, the good old parabola.0326

Consider if we wanted to find what value x2 approaches as x approaches 2.0332

As x gets closer and closer to 2, what does x2 become?0337

Well, if we graph f(x) = x2, we see that the answer is exactly what we would expect.0342

2: as we get really close to a 2, we end up getting really close to a 4.0349

As we come in from the left side, we can see that the value we are getting to, as we get closer and closer0356

to a 2 from the left side--our vertical height gets closer and closer to a 4.0363

Similarly, if we come in from the right side, as we get closer and closer to 2, and x = 2, from the right side,0370

on our horizontal location, we get closer and closer to a vertical location of 4.0377

We end up getting 4 as the limit: as x approaches 2, x2 gets really close to 4.0382

It becomes 4 as x approaches infinitely close to 2.0388

So, as x gets closer and closer to a value of 2, f(x) gets closer and closer to a value of 4.0393

And that is why we end up getting this limit.0402

Now, to expand our ideas, let's modify the function and see what happens.0405

Consider the piecewise function g(x), which equals x2 when x is not equal to 20409

(that is most of our parabola right here) and 1 when x equals 2.0418

At the specific value of x = 2, as opposed to following our normal parabolic arc, we end up putting out just a value of 1.0424

This is a piecewise function; if this is not sounding familiar, and you have no idea how to interpret this sort of thing,0433

go back to the lesson...we saw this a long time ago, near the beginning of this course...the lesson on Piecewise Functions.0439

They will show up a lot, especially in the beginning of calculus.0444

They are an important thing to have in your understanding.0447

So, if this doesn't make any sense, go and check out piecewise functions from the early part of the course, when we were studying functions.0450

All right, with this idea in mind, let's ask what the limit is, as x approaches 2, of g(x).0456

Well, what we want to know--we might be tempted, first, to say, "Look, it is right here; that is what x is at 2."0463

Yes, that is what g(x) is at 2; g(2) is 1.0469

Well, yes, but we can also think about it as what we are getting close to.0476

Well, as we get close to x = 2 from the left side, we see that we are getting really, really close to what height?--the height of 4.0482

As we approach from the right side, we see that we are getting really, really close to what height?--the height of 4.0492

So, our first automatic response might be to think that it has to be here, because that is what it is at x = 2.0499

But the limit isn't about where you actually are; as a limit, x/x, as x goes to 0, produces 1,0505

even though actually the function just fails to produce anything.0516

A limit is about where you are headed towards, not what actually happens at that location.0519

We don't really care about what happens here; this part isn't important.0525

The thing that actually happens at 2 isn't important; it is a question of what happens on our way to 2.0529

Well, on our way to 2, the thing that we are getting close to is this location at 4.0535

That is the location we are getting to; so the answer for this limit will be 4.0541

So, g(2) is not equal to 4; g(2) is equal to 1, because at x = 2, we just put out 1 for that function.0547

But as x approaches 2, the value of g(x) approaches 4.0555

g(x) jumps; it does this sudden leap off of the parabolic arc, only at x = 2; it only swaps at x = 2.0561

What it seems to do, up until that moment, is behave like x2, because for everything that isn't x = 2,0571

for everything other than x = 2, it behaves just like x2.0579

Up until that moment, it is behaving just like x2; and because of this, it has the above limit of becoming a 4.0583

It is going towards a 4 until that single moment where all of a sudden, when it actually touches the 2, it jumps away.0590

But up until that moment, it seems like it is going there; so that ends up being our location.0596

The vertical height that it seems to be going to as x approaches 2 is 4.0601

In a way, we can visualize what is going on by doing the following:0608

Begin by graphing the functions normally; we graph f(x) normally, x/x...oops, that should not say x/x;0611

that should be x2; I'm sorry about that; that should be f(x) = x2.0618

And over here, g(x) = x2 when x is not equal to 2, 1 when x equals 2.0624

We have this single part that jumps away--this single point that is not on the normal curve.0630

Then, what we can do is end up covering up the part that we are going to.0634

Notice: in this case, what we are about to consider is the limit, as x approaches 2,0640

as we get close to this value, as we get really close to this on both of them.0645

With that idea in mind (I'm going to swap this back out; it should be an x2; we are looking at the limit as x goes to 2),0651

as we take the limit, what we do is cover up the horizontal location that x is approaching,0661

because, since it is a limit, what we are concerned with is what happens on our way to that value.0668

But we don't actually care about that value in specific.0673

We don't care about that horizontal location; what we care about is our way to the horizontal location.0676

It is the journey that matters, not the destination, when it comes to limits.0682

So, the limit is the height we expect--what we feel like would happen without peeking under the cover.0686

We have that black bar there that keeps us from being able to see what it actually turns out to be.0694

But in this case, it seems like what it is going to come out to be is 4.0699

If all of the information we have is just the picture in front of us (except that part that has been covered up--0703

we are not allowed to look under it), the information that we have makes it seem like it looks like it is going to 4 in both of them.0707

The idea is the question of what we expect will end up happening.0713

Where does it seem like this function is going to? That is what a limit is about.0717

With all of that in mind, we now have the ability to create a more formalized definition.0722

The definition of a limit: if f(x) becomes arbitrarily close to some number l, as x approaches some number c,0727

but is not equal to c (we don't actually care about that horizontal location;0735

we just care about the way to that horizontal location), then the limit of f(x), as x approaches c, is l.0739

Symbolically, we write that as "limit as x goes to c of f(x) equals l"; the limit of f(x) as x approaches c is l.0747

So, as x gets close to c, what value does our f(x) seem to go to--what value is f(x) getting towards as we get close to that horizontal location?0757

There are two important things to note: we are looking at f(x) as x goes to c, but are not concerned with x = c.0769

So, for the purposes of a limit, x is never equal to c; we are only concerned with what happens on the way to c, but not actually x at c.0777

Remember: it is the journey, not the destination, that matters when it is a limit.0786

The other thing to notice is that, when we consider x approaching c, we are considering x approaching from all directions--not just one side.0790

It is not just this side; it is not just this side; it is both sides coming together.0798

To have a limit, f(x) must go to the same value from both sides.0804

The right side and the left side have to agree with each other.0809

If they go to totally different things for a horizontal location, if they go to totally different heights, then they don't agree; there is not a limit.0812

There is not a sense of what to expect if they are going to totally different places.0819

Where is it going to be? Is it going to be somewhere in the middle?0824

Is it going to be the top one? Is it going to be the bottom one?0825

We don't have a good sense of which side to trust, so we can't get a limit out of it.0827

The two sides have to agree for us to end up having a limit.0832

Technically, I want to point out that this isn't actually the formal definition of a limit.0836

That said, what we are seeing here is going to certainly be enough for now.0841

This idea, the definition of a limit that we just talked about (this whole thing here and all of our commentary that came after it--0846

all of the stuff that we have been working through so far in this lesson)--this is plenty for the class you are currently in.0852

A precalculus-level course like we are in right now--this is more than enough of an understanding of what a limit is.0857

You are doing great at this point.0862

Even for a calculus-level class, this is really pretty much enough understanding.0864

Some courses will maybe vaguely talk about the formal definition,0868

but very, very few will really expect you to fully understand the formal definition of a limit.0872

This is really all that they are looking for--this sense of a limit being what you are going towards, but not where you actually end up.0877

And for pretty much most science courses, most engineering courses, this is really all you need.0885

If you want to talk about the really formal definition of a limit, that is going to show up later on,0889

in really advanced math courses, like second--maybe even third-year college math courses, really proof-heavy math stuff.0894

And I think that that is really great stuff.0901

But for the most part, you will be fine with just this, probably forever.0903

However, if you are really interested in mathematics, if you are curious, you might want to check out the next lesson, Formal Definition of a Limit.0907

I think that this stuff is really, really cool; and I have a great lesson that will help us explain and understand what the formal definition of a limit is.0913

But frankly, probably 99 times out of 100, you are never going to need to know that stuff.0920

Pretty much any class that you will be taking in the next two years is never going to actually require you to know the formal definition of a limit.0924

So, don't worry too much if you don't feel like watching it; it is totally a fine lesson to end up skipping.0930

But if you are interested, it is really cool; and if you are interested in this, you might end up being interested in taking advanced math classes later.0935

And it will totally come into play later on, and you will be a step ahead of everybody else in understanding this fairly complicated idea.0940

All right, let's keep talking about limits: so far, all of the examples we have seen have had limits.0946

But a limit does not always exist: for example, consider the limit as x goes to 0 of 1/x.0952

Well, here what we are doing is looking at as x goes to 0 (the y-axis is at x = 0).0958

As we come in from the right side, we aren't actually going to a single value.0965

We are just going to go up and up and up and up and up and up and up and up, and we don't ever stop going up.0969

It is a vertical asymptotes; when we worked on rational functions and vertical asymptotes, we just went on up forever.0975

There is no single l value, no single limit number that we are going towards.0980

So, there is nothing that can be agreed upon.0985

And even worse than that, we end up going in the totally opposite direction when we come from the left side.0987

One side is shooting off to positive infinity; the other side is shooting down to negative infinity.0991

There is nothing to say for the limit here.0996

They are not approaching something where they are agreeing on some value.0997

They are not even going to separate values; they are just blasting off to infinity on both sides.1001

We don't really have a good way to talk about this; there is no way to assign a specific number value that we expect will happen at x = 0,1005

because it is just going to clearly go crazy, and that is that; there is nothing that it is going to go to.1012

So, because of that, we say that the limit does not exist.1016

This limit does not exist; there can be no limit, as x goes to 0, because there is no single value that 1/x is headed towards.1019

There is nothing that we are going to end up seeing them agree on.1028

Even one side is not going to agree on anything, because it just keeps going up.1032

There is no single value; therefore, there is no limit that we get out of it.1035

Still, I want to point out: for x approaching any other value than 0, the limit would exist, because it would approach a single value.1039

If we approached 1 as our thing, then we would end up approaching 1.1048

If we approached 4 as our thing, then we would end up approaching 1/4.1053

If we went to -3, if we were approaching -3 from both sides, then we would be going to -1/3.1059

All of those make sense; the only issue is here at x = 0, where, because it has an asymptotic thing, it just goes crazy and shoots off in both directions.1066

There is nothing that we can end up pinning it down with, so we have to say that the limit does not exist.1073

But anywhere else on 1/x would exist; but x going to 0 does not exist for 1/x.1077

In the previous example, the limit as x goes to 0 of 1/x, that didn't exist.1084

But at the same time, 1 divided by 0 does not exist, either; f(0) didn't exist, and so limit as x goes to 0 of f(x) didn't exist.1089

And there is a connection there; however, it is possible for the function to exist while the limit does not exist.1097

So, to see this, take, for example, g(x) equals the piecewise function x2 when x is less than 1.1105

(we see this parabola on the left side) and 4 - x when x is greater than or equal to 1 (and we see this straight line on the right side).1112

We have every single point in the real numbers defined.1121

They will end up having some value that the function will put out.1124

If you plug in any number, it is going to either be on the straight line portion, or it is going to be on the parabola portion.1128

So, g(1) = 3; if we plug in 1, we use 4 - x; 4 - 1 gets us 3; that is this point right here.1134

However, the limit as x goes to 1 of g(x) does not exist; why?1148

Well, from the left side, we end up approaching this value here.1153

We are approaching this side as we approach from the left side.1157

However, as we approach from the right side, we end up approaching this totally different value.1160

And so, there is this big gulf between the two limits.1166

We are going to two totally different places at this horizontal location.1169

So, the function approaches two totally different values from the right and left sides.1173

Since they don't agree, we can't say, "Oh, that is what we expect," because we have totally different expectations from the two sides.1180

So, that means that the limit does not exist.1186

How do we actually go about finding limits in general?1190

The first really great way to do this is with graphs.1192

One way to find the value of a limit is just to look at a graph of the function.1195

If you have a graphing calculator, you can plot it on the graphing calculator.1198

If you have some sort of graphing program, like I make these graphs with, you can plot it on a graphing program.1201

Or if you are just really good at making graphs, you can draw a graph.1205

Figure out if a limit there makes sense: is there something where both sides are going to the same thing?1208

And if so, find what value the graph indicates.1214

For this case, we have f(x) = (x2 + x - 2)/(x2 - x);1217

(x2 + x - 2) on the top, (x2 - x) on the bottom of the fraction.1223

So, we see that there are some parts where we have issues.1227

If we plug in x = 0, we end up having an asymptote; and if we plug in x = 1, we end up having this hole here.1230

But we can still ask what the limit is as x goes to 1.1237

Well, if we go to the graph, as x goes to 1 from the left side, we end up seeing that we are approaching that height of 3.1241

As x goes to 1 from the right side, we see that we are approaching the exact same height.1249

We end up getting up a value of 3; either way we approach this, we are going to end up seeing that 3 is what the expected value is.1255

The graph shows us that we are working towards 3, or something at least on this graph looks very close to 3.1263

So, we could say 3; but of course, we are reading a graph.1267

Reading a graph isn't always as precise as we would like.1271

Reading a graph, we know that you can sometimes be off by 1/2 or 1 whole thing.1275

So, it doesn't give us a perfect answer; but it gives a pretty good idea.1279

We have a good sense of what it is going to be, although it is not perfectly precise.1283

However, limits do have the massive benefit of being able to allow us to get an intuitive sense of how the thing is working.1287

They will let us see what the function is, as a general idea; and sometimes that is the most useful thing of all.1293

Graphs are really, really handy, even if they don't let us see precisely what the value is;1297

they let us understand what is going on--does it even make sense for it to have a limit here?1301

Things like that are what a graph allows us to answer.1305

Alternatively, if we want something that is more precise, if we want a more precise sense of where the limit will go,1309

or we don't want to graph the function, just because we have some sense of what the graph looks like,1315

or we just don't feel like graphing it, because we know it is going to be a pain,1318

but we have enough of an idea to know that the limit would exist there; we can use a table of values,1320

where x will approach the value that it approaches in the limit.1326

Once again, we have the same f(x) = x2 + x - 2 over x2 - x.1331

And now, we are looking at the limit as x goes to 1 of that function, the same limit as before.1335

What we can do is: we have 0.9, 0.99, 0.999...we are approaching 1 from the left side there.1341

We can approach it from the right side: 1.1, 1.01, 1.001...we are getting closer and closer values.1350

Now, of course, we can't actually plug in x = 1, because if we plug in x = 1, it is just going to fail on us.1357

The limit isn't about where it actually would be; it is about what happens on the way there.1364

So, we don't plug x = 1 into our table; all we are concerned about is what happens to the numbers as they get really close to x = 1.1368

We calculate this with a calculator: .9 comes out to be 3.222; .99 comes out to be 3.020; .999 comes out to be .002.1375

Going from the other side, 1.1 is 2.818; 1.01 is 2.980; 1.001 is 2.998.1385

So, we can see that the value that we are getting close to, 2.998, 3.002...we are clearly tending pretty close to the value of 3.1393

So, we can assign that this limit is going to end up having a 3 as we get closer and closer and closer.1403

Now, maybe we are off by .0001 or some small number; but we can be pretty sure that,1409

unless it does some really sudden jump there, 3 is probably going to be pretty close to it.1416

That lets us get a good approximation--probably a good approximation within many decimal places, but we are not absolutely, precisely sure.1421

Still, that is really, really close; and the closer our table has x approach the limit, the more sure we can be.1429

If we, instead of using 1.001, would use 1.0000000001, we would be that many more decimal places sure of where we are headed.1438

The same happens with 0.999999999; by plugging in more and more decimal places, we get more and more accuracy.1448

So, we can be more and more sure of what the value that we will end up getting out of that limit is.1456

If you have a graphing calculator, this is a great use for the table of values feature,1460

where you can just set up a function, then go the table of values,1466

and have it be an independent thing where you plug in each number.1469

You plug in .9, .99, .99999; you hit them all in, and it will just put out values for you.1471

And you won't have to type in the entire expression over and over.1476

So, check out the course appendix; there is that appendix on graphing calculators as part of this.1480

It is a great thing to check out if you have access to a graphing calculator.1485

That table of values will make your life so much easier when you are working through this sort of thing--this is a really good use here.1487

Precise methods: the lesson after the next is finding limits (not the very next lesson,1494

which will be the formal definition of limits, which you can totally skip, if you are not particularly interested;1499

but the one after that--you will want to watch it, and that is Finding Limits).1504

In it, we will see ways to precisely find the limit of a function through algebraic methods.1508

Graphing and a table of values--those two things give us really good approximations.1512

They give us a good sense of what is going on, but they don't tell us what the value has to be, precisely.1517

We will figure out algebraic methods in the next, next lesson, Finding Limits.1521

For right now, though, we will just stick to the methods of graphing and making tables: they are pretty good methods, actually.1526

Even after you learn those more precise methods about how to figure out algebraically, precisely, don't forget about these.1532

They can be really handy when you can't figure out how to figure it out precisely, algebraically,1538

but you still need to evaluate it and get a good sense of where it is going.1543

You can always use these methods.1546

Now, often you will end up having problems where the problem says that you have to get it precisely, and you won't be able to use these methods.1549

But sometimes, you will have some really, really complicated thing, and you won't be able to work it out.1555

You can just put it in a table of values, something like this, and you will be able to get a good sense of where it is headed; and that can be useful.1559

All right, we are ready for some examples.1566

For each limit below, if it exists, determine the value using the associated graph.1568

Our first one is the limit as x goes to 0 of 1/x2.1571

Well, what we can see is that, as we get closer and closer to x = 0, this thing just shoots off; both sides fly off to infinity.1576

They shoot off forever and ever and ever.1585

So, even though they are both going towards positive infinity, do they ever end up establishing at a single value?1587

Does it ever settle on a single value? No, they are always going to keep going up.1593

It is going up asymptotically, so there is never a number that they get steady out on.1598

They never decide to land on 47 or some other number.1603

Since they keep going on forever, it doesn't have a limit.1606

The limit here does not exist, because it never settles on a single value.1609

It shoots up to infinity forever, and since it never ends up staying around at a single value, it never settles on a single value.1619

So, we end up not being able to get a limit out of it.1628

The next one is the limit as x goes to -3 of (x + 3)/(x2 + 5x + 6).1632

In this one, if we plug in -3, look: there is a hole, so we can't just directly plug in -3.1639

But from the graph, we can see that, yes, there is nothing wrong.1645

We are approaching the same thing from the left and the right side.1648

What value are we approaching? We are approaching a value of -1, so the limit is -1; great.1651

The next one: Evaluate the limit as x goes to -2 of 3x2 + 5x + 1.1659

We could graph this; this is just a parabola--it wouldn't be too tough for us to graph.1664

And if we graphed it, we would end up seeing that it looks something like this.1669

And we could figure out what it is--draw a really careful graph; but drawing a really, really careful graph takes a fair bit of effort.1676

And we have a good sense that there are going to definitely be limits everywhere,1682

because it never does anything weird; it never jumps around; everything that we expect to happen is what happens.1685

So, we don't have to worry about that; but that is what the graph gets us.1690

At this point, now we want to actually figure out what the value is.1693

The easiest way to figure out what the value is: we can just plug in values; we make a table of values.1695

We are going to have x on the left side, and the f(x) that comes out on the right side; what value are we approaching?1702

We are approaching -2; so x going to -2...if we are a little below -2, we will be at -2.1, then -2.01, then -2.001.1708

On the other side, we will be coming away from -2: -1.999, -1.99, -1.9...1722

If we plug these things into our calculator, -2.1 gets us 3.73; -2.01 gets us 3.0703; -2.001 gets us 3.007003.1732

Flipping to the other side, -1.9 would get us 2.3; -1.99 would get us 2.9303; -1.999 gets us 2.993003.1748

So, it is pretty clear what we are approaching.1763

As we go in from each side, -2.001, -1.999...they are getting really, really close to this middle value of 3.1765

So, that is what the limit ends up being; the limit comes out to be equal to 3; that is the value that it is approaching.1774

Also, remember how we talked about...well, look at that graph: the way that that graph is there--it doesn't do anything weird.1781

Everything that we expect to come out of it is what is going to be that function's location.1789

So, another thing that we could do is: because, in this specific case, the graph doesn't do anything weird,1794

the function doesn't jump around in any weird ways; there are no breaks; and there is nothing strange about it.1799

So, if there is nothing strange about it, what we could do is: we could also say,1806

"Well, that means that the limit has to be what the function actually ends up going to at that point."1808

So, we could also just evaluate it by plugging it in: 3(-2)2, plus 5 times -2, plus 1.1813

3 times...-2 squared gets us 4, plus 5 times -2 gets us -10, plus 1...3 times 4 is 12, plus -10...so 2 + 1...1821

That came out to be 3 as well, so that checks out, as well.1830

So, what we just saw there was actually one of the precise methods of doing this stuff.1834

We will talk about this more in Finding Limits; but it seemed like a really easy one for us to see...1838

we are going to start to get a sense of how this stuff works, and we want to find the precise stuff.1842

So, we will see more, and we will also understand exactly why we can do this, in the coming lessons.1846

All right, the third example is the limit as x goes to 0 of sin(x)/x.1851

If we were to graph this, if we used a graphing calculator to graph what comes out of this, we would see that it is going to look something like this.1856

Oops, I actually made a mistake with that graph; we would see that it was going to look something like...there we go; that is better.1865

It is going to look something like that.1877

The easiest way to do this is to just do a table of values; we have x, and f(x) is coming out of it, as we plug in various values for x.1880

So, our x is approaching 0; if we are approaching 0 from under it, we are going to be at -0.1, then -0.01, then -0.001.1890

And you could use different numbers, as long as they continue to get closer and closer to 0.1902

But I think those are pretty easy ones to use.1905

From the other side, we would be coming away: 0.01 on the positive side now; 0.01, and 0.1, now that we are past the 0.1908

We plug these into a calculator; we figure out what it is; we get...-0.1 going in gets us 0 point...1917

Also, remember: this has to be in radians for us to...1922

Actually, it doesn't have to be in radians because of this specific problem.1925

But any time we end up seeing a function, we should assume that it is in radians,1927

unless we have been explicitly told otherwise--that it is in degrees.1931

But normally, assume that it is in radians when you are doing math.1934

0.998334...plug in -0.01: 0.999983; -0.001: 0.9999999.1938

On the other side, there is 0.1: we have 0.998334; for 0.001, it is 0.999983; and for 0.001, 0.999999.1957

It is pretty clear what we are ending up approaching here.1976

As we get closer and closer, the value that we are approaching is 1.1978

The value that this is getting really close to is 1, and it seems to show that it is going to get really close to it.1981

So, we see that the value of this limit is 3.1988

The fourth example: Using the associated graph, explain why the limit below does not exist: the limit as x goes to 0 of sin(1/x).1992

Just glancing at this thing, the answer is simply that it goes crazy.2002

Look at this thing that is happening here--it is going crazy!2009

This graph is doing some really weird stuff as x gets close to 0.2014

So, as x gets close to 0, what is going on?2019

Well, notice how we can see that it is going up, and then it goes all the way down.2021

And then, it goes all the way up, and then it goes all the way down.2026

And then it goes all the way up and all the way down; the same thing happens on the other side:2029

up and then down, and up and then down, and then up and then down.2032

Well, what it seems to be doing is going up/down, up/down, up/down, faster and faster, as it gets closer and closer.2035

It is going crazy; it is bouncing up and down forever as it gets closer to this x going to 0.2040

It is just bouncing so, so fast; so as we get closer and closer to that x going to 0, we don't have any idea what to say,2053

because we are flying around; the closer you get to x = 0...it constantly changes.2060

The function is constantly changing, constantly going up and down.2065

Because it is constantly bouncing up and down forever and ever and ever and ever, we end up saying that it doesn't have a limit.2068

It doesn't exist; the limit does not exist, because it doesn't settle on anything.2074

A limit has to be settling towards some value, and it has this craziness there.2078

So, we end up not being able to say it has a limit; the limit does not exist.2083

If you want a better idea of what is going on here (and I think it is always cool to have a better idea of what is going on),2088

we can break this into two pieces: the graph of 1/x ends up graphing (let me put that in black,2092

just so we can easily see it)...here is our axis...it ends up looking like this.2099

We are used to that nice vertical asymptote there.2110

And then, the graph of sin...let's say t, just some dummy variable t that we plug in...it is going to end up having that nice periodic graph.2112

That is how sine works: it goes up and down and up and down and up and down and up and down.2124

That is what happens; however, what we have here, the t that is going into this, is 1/x.2130

So, what happens is that the value, as we get closer and closer to 0...2138

well, what happens to that asymptote as we get closer and closer to 0?2144

As we get closer and closer to 0, they shoot off to infinity.2148

So, what we have is: we effectively have an infinitely long amount of stuff that we are plugging into sin(t) in a very small, compact space.2151

That is why we end up seeing that it is going slowly.2160

And then, as it gets closer and closer to 0, 1/0 (that is not really formally accurate, but)...1/0 is effectively shooting off to infinity.2163

1/0 shoots off to infinity; so if we are shooting off to infinity, and then we are plugging something2173

that is going all the way out to infinity into sine, well, that means our periodic thing is going to go up and down forever.2178

But because we are doing all of this forever-ness in all of this very tiny distance of .1 to 0,2184

that means we end up having it go faster and faster and faster and faster, because it has to manage2190

to do all of infinity in 0 to 0.1--pretty crazy.2194

That is why we end up seeing this behavior.2200

And if that still doesn't quite make sense, don't worry.2202

Honestly, you will be fine in a precalculus class, and even a calculus class, without fully understanding this idea.2204

But if you let this rattle around in your head--think about it for a while--you might start to think, "Oh, I am seeing it."2209

There are some really cool ideas in math, and lots of the cool ideas in math end up taking a little while to fully understand.2215

So, even things that you don't understand the first time--the second time you look at it, it might make a lot more sense.2220

I think this stuff is really cool.2224

All right, we are ready for our final example.2225

For each limit below, if it exists, determine the value using the associated graph.2228

Our first one is limit as x goes to 0 of 2x/x; well, that one is pretty easy.2232

It is clearly going towards 2, because it is just in a steady state at 2 all the time.2237

While actually plugging in 0...if we plug in 0, 2(0)/0 gets us 0/0; we can't plug in 0.2242

So, it doesn't exist at 0/0; but on the way to that x = 0, it exists just fine.2248

We can see what the limit that it is going towards is.2254

The limit that it is going towards is 2.2257

Similarly, over here, the limit as x goes to -2 of (-3x - 6)/(x + 2): what is it going towards?2260

It is going towards that thing, which is -3; so it is going to -3.2267

If we actually plugged -2 in, well, x + 2, so -2 + 2, would get us dividing by 0; things would fall apart; we are not allowed to do that.2272

But as long as we are not plugging in -2 directly, everything is fine.2286

We always end up having -3, as we can see from this picture right here.2291

So, since it is always -3, then the limit of what it is going towards (we don't have to worry about that thing that it is actually at) is -3--as simple as that.2295

Also, I want to point out to you something that we are going to see in the finding of the values of limits.2305

We can actually get an early start on what is going to happen, to get that idea percolating through your head.2309

Look: 2x/x...well, at x = 0, it ends up doing something different.2313

But with the exception of x = 0, 2x/x behaves exactly like 2.2319

So, because it behaves exactly like 2, we can effectively say, "Well, what would it be at 0, if we were using this other alternative way of talking about it?"2325

The other alternative is that you are always 2; so since it is always 2, we end up getting a limit of 2 out of it.2334

The same thing is going on over here with (-3x - 6)/(x + 2).2340

Well, we can rewrite that top as -3 times (x + 2) over (x + 2), which means we can then write an equivalent thing,2349

with the exception of x = -2; everywhere else, we won't have that issue.2358

It would just be the same as -3 forever and always.2363

So, with the exception of that x = -2, it works just fine.2367

But we don't actually care about -2; remember, there is that idea of blacking out, where you are covering up that chunk,2370

because we are not allowed to peek underneath the cover.2379

So, if we cover up that chunk, and then we try to figure out where it is headed towards,2382

we don't have to worry about the fact that, if we had plugged in -2 here, it wouldn't work,2388

because we are not worried about what happens at -2.2392

We are just worried about what happens everywhere else; and what it is equivalent to everywhere else is -3.2394

And so, that is why we end up seeing it.2400

This is just a quick preview of the ideas we will end up getting when we start talking about finding limits precisely with more algebraic methods.2401

All right, that gets us a really good idea of how limits work.2407

Just remember: it is the idea of where you are going and if it matches from both sides;2410

the question is of where you are going, but it doesn't matter what it actually is there.2414

It is about the journey, not the destination.2417

All right, we will see you at Educator.com later--goodbye!2420

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