Home » Mathematics » Math Analysis » Formal Definition of a Limit

Vincent Selhorst-Jones

Vincent Selhorst-Jones

Formal Definition of a Limit

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Table of Contents

Section 1: Introduction
Introduction to Math Analysis

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 (7)

1 answer

Last reply by: Professor Selhorst-Jones
Mon Aug 3, 2015 5:54 PM

Post by Mike Petersen on July 3, 2015

Would this be considered the "squeeze theorem"?

1 answer

Last reply by: Professor Selhorst-Jones
Mon Jul 21, 2014 8:29 PM

Post by Teemu S on July 20, 2014

What an awesome lecture! I know I'm going have to deal with this definition in a couple of months and this has cleared it up for me.

I really appreciate all the lectures you've given. You are by far the most motivating and inspiring math teacher I've come across. A big thank you from Finland!

2 answers

Last reply by: Jamal Tischler
Sun Mar 1, 2015 7:09 AM

Post by Jason Todd on July 27, 2013

Awesome Video!!! It grabbed my intuition and slammed it into synchrony with this concept. Thank you! The dialogue made things really enjoyable.

P.S. f(x)= x sin(1/x) is my new favorite function! Do you have a list of really cool functions like this?

Formal Definition of a Limit


    Note: Very few students will have any use for what is in this lesson, let alone during a pre-calculus level course. Problems based on the formal definition of a limit are extremely rare in calculus class. This stuff won't come up in science classes, and will only be necessary in high-level college math. Still, if you're interested in math for math's sake or know that you want to one day study higher level mathematics, check this lesson out. It's okay if it doesn't all make perfect sense today, but seeing these ideas now will help you down the road when it starts mattering. Plus, it's fascinating stuff!
  • We need two new greek letters:    Delta⇒ δ    Epsilon⇒ ε
  • Formal Definition of a Limit: Let f be a function defined on some interval (a,b) of the real numbers, where a < b. Let c ∈ (a,b) and L be a real number. Then,

    lim
     
    		f(x)   =  L
    means that for any real ε > 0, there exists some real δ > 0 such that for all x where 0 < |x−c| < δ, we have |f(x) − L| < ε.
  • The first half of the definition sets up the groundwork. The function is defined, there is a horizontal location we're approaching (c), and some vertical location the function goes towards (L).
  • The second half of the definition is the difficult part. It says that for any ε > 0 boundary around L, we can restrict our x to within some δ > 0 boundary of c to force f(x) to stay within the ε-boundary around L.
  • Check out the video to see some diagrams and more explanation of this idea. It's a confusing idea at first, and it takes awhile to make sense of it.
  • We can imagine a limit as a sort of never-ending debate between two people. One person gives hypothetical ε-boundaries, while the other person has to defend by giving a δ-boundary that would keep the function within the ε-boundary.
  • To formally prove a limit exists, we must show that for any ε > 0, there will always be some δ > 0. This normally takes the form of creating a formula for δ based off ε, because if we can create such a formula, then we will have shown δ will exist no matter what ε is chosen.

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

Transcription: Formal Definition of a Limit

Hi--welcome back to Educator.com.0000

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

In this lesson, we will explore the formal definition of a limit.0005

Previously, we defined a limit with intuitive terms, like "approach" and "goes to."0007

Now, we are getting serious, and it is time for some really technical stuff.0012

However, I want to point out that very few students will have any use for what is in this lesson, let alone during a precalculus-level course.0015

I want you to have the option to know the formal definition, if you are interested.0025

But most people will not need it, even in math class.0028

Problems based on the formal definition of a limit are extremely rare in calculus class--maybe as a bonus question, but probably no more.0031

This stuff won't come up in science classes, and it will only be necessary for high-level college math.0040

And when I say "high-level college math," I mean that this stuff is really going to only definitely start showing up0047

by the second or third year of taking advanced college math courses.0051

This stuff is not going to show up any time soon, so I really just want to be frank with you and honest:0055

what I am going to teach here is really a totally optional lesson.0060

It is not going to come up in the course that you are currently taking; I guarantee you that.0064

And it may or may not come up in the next course you take, if you go on to take calculus.0068

But it is really, really unlikely to be tested heavily.0072

It will maybe be a bonus question, or maybe one real question; but for the most part, you are not going to see this formal limit definition.0075

It just doesn't show up; it has very little direct applicability in the sciences; it is only used for some pretty advanced ideas in mathematics.0083

It is really important for building advanced ideas in mathematics.0091

But for day-to-day use of mathematics, we just never really need to see it.0094

Unless you are particularly interested in mathematics, or you know that you really want to go into high-level physics,0098

or you are interested in something like computer programming, where you will have to get enough mathematics learned,0103

at some point, that you will end up needing to be able to understand those advanced math concepts,0107

you are probably never going to end up encountering this formal definition, ever.0111

And that is totally OK; wanting to learn this sort of thing--I know that it is not for everybody.0115

But for me, I personally love this stuff; this is the reason I love math--because there are these really cool, complex,0119

strange, puzzling logic ideas that make this really interesting structure that is just fascinating--to me, at least.0127

So, I think this stuff is totally awesome.0133

If you are not interested, don't worry about it; skip this lesson.0136

There is going to be some really difficult stuff, and there is not really a whole lot of direct personal need to understand it.0139

But at the same time, there is not really a whole lot of direct personal need to go to a museum; you do it because you are interested in the art.0144

And that is the same reason that you might be interested in mathematics.0150

It is the reason I am interested in mathematics--just because I find it personally satisfying and cool.0153

If you are interested, let's go and check some stuff out.0157

OK, and before we keep going, I just want to say, for those of you who decided to stay around, thanks.0162

I think this stuff is really, really cool; I am glad to get the chance to share it with you.0166

And I think it is really, really cool; I hope you end up liking it as much as I do.0170

We few, we happy few, we mathematicians--we get to see this.0175

All right, we are going to need some new Greek letters before we get started.0180

That is right: this stuff is so cool that we need to bring out the new Greek letters.0183

Our first one is delta, and these are the lowercase versions; so δ is lowercase delta, and ε is lowercase epsilon.0187

Here is δ; and if you are going to draw it by hand, it is kind of hard to draw this thing exactly as it ends up getting typeset.0194

So, I would recommend...you want something on the top, and then sort of a curved loop like that.0201

You might end up seeing me write it like that, as I am writing it pretty quickly.0206

But that is a pretty good idea of what you are trying for.0209

This one is kind of not a very good one; but put some sort of curved loop on top, and then a sort of circular thing on the bottom.0211

And that will be enough for most people who understand mathematics to realize that you are trying to write delta.0217

You don't have to ahve it absolutely perfect, because they will know what the symbol is, especially when they end up seeing it a bunch of times.0221

Epsilon is considerably easier to draw; it is just sort of a backwards 3; that is it.0226

That is ε; that one is easier than δ, I would have to say.0231

All right, the formal definition that we are about to see involves these letters.0233

So, it is sometimes called the (ε,δ)-definition of a limit, or just the formal definition of a limit.0238

We just talked about the intuitive idea of what a limit is; and that is a really great way to think about limits.0244

But sometimes you really need to get to the basic facts and turn intuitive ideas into precise things, so that you can say "this is what it is."0249

And that is what this definition is for.0256

All right, let's get to it: the formal definition of a limit.0257

Let f be a function defined on some interval a to b of the real numbers, where a is less than b.0261

Let c be contained in a to b, and let l be a real number.0268

Then, the limit, as x goes to c, of f(x) = l means that, for any real ε greater than 0,0274

there exists some δ greater than 0 such that, for all x where 0 is less than the absolute value of x - c,0282

which is less than δ, we have the absolute value of f(x) minus l is less than ε.0289

What? Don't worry: if you found that definition confusing, that is exactly how I felt when I first heard it.0296

The first time you see this, you think, "What is going on?"0304

That is totally OK; it is something that you pick up and work through slowly, over time.0307

If you opened a book for the very first time, you wouldn't expect to be able to understand every word on the page.0311

It is something you have to work through; it is OK--you are building up these ideas of understanding it.0316

It is not something that is going to make sense exactly the first time.0319

It is something that makes sense over time; that is totally fine.0322

So, we will start working through it, step-by-step, and we will see some ways to understand what is going on in this definition--0325

to see how we can parse this thing and make sense of it.0330

All right, first, let's get the basics out of the way, the ground work here.0333

The first half of the definition is just setting up the limit; that is what we end up getting in this first half.0337

Let f be a function defined on some interval a to b (they are real numbers) where a is less than b, and c is contained in a to b, and l is some real number.0342

All right, let's break that down, piece by piece.0352

First, let f be a function defined on some interval a to b.0354

That just means that it is a function, and it is defined over some portion of the reals.0358

There is some piece of it that it ends up working in.0363

It is basically just saying that this function is defined somewhere; that is all that we are saying.0366

There is some chunk of the reals where this function makes sense.0370

This part where we say a < b is just a guarantee that we don't accidentally say a = b, and we end up having something that doesn't make sense.0373

It is so that we know that we are going from some a, up until some b; that is our interval of where the function is defined.0379

Next, the horizontal location c, contained inside of (a,b): remember, this symbol just means "inside of."0386

So, (a,b) is a set of all possible values; it is the interval, and that symbol just means that c is inside of that set of possible values.0393

So, the horizontal location c contained inside of (a,b) is the value that the limit will approach.0400

That is the thing where we are getting closer and closer to that, and we are seeing where we end up going to.0406

So, c is just a location that we are getting towards; that is the value that the limit will approach.0412

And finally, l is some real number; l's real number is the vertical location that we end up going to.0417

That is where the limit will go to; we are setting the l that we are saying our limit ends up going to as we approach this c.0424

So, that is the basics that we are setting up: f is some function; it works on some chunk.0432

c is where we are going to, and l is the value that we end up meeting up at, that we are headed towards.0436

All right, the hard part is the second half of the definition.0443

The limit as x goes to c of f(x) = l means that, for any real ε greater than 0, there exists some real δ greater than 0,0447

such that for all x where 0 < |x - c| < δ, we have f(x) - l < ε; OK.0455

To break this up, we want to be able to focus on this one piece at a time.0462

So, what we are going to do is focus on the ε portion first.0465

It will be easier to understand that; and then we will see the δ part in that idea.0468

So, for any real ε greater than 0: what that means is that, if we just grab some ε--0473

we just say some number, like 1 or 1/2 or whatever number we feel like saying,0479

we will end up having this be true later on: that the absolute value of f(x) - l is less than ε.0483

Now, that might be kind of hard to understand: the absolute value of f(x) - l is less than ε...0490

we are not used to working with absolute value that much; it is kind of confusing.0493

But remember one of the things we talked about when we first talked about absolute values.0496

Absolute value is a way of seeing the distance between two things.0499

If we have the absolute value of m - n, if we have the absolute value of q - r, that is a way of saying the distance between those two points.0502

The absolute value of some object minus some other object is just a way of saying how far those two objects are from each other.0512

That is one of the ideas that we have from absolute value.0519

When we say the absolute value of f(x) - l, what we are really saying is how far f(x) and l are from each other.0522

Now, that means, with the combination of ε being greater than 0, and the distance between f(x) and l being less than ε...0529

what we are doing is saying that there is this boundary around l.0537

The maximum distance f(x) can be from l, the distance between f(x) and l, has to be less than ε, this thing right here.0540

Since the distance between our f(x) and our l is less than ε, it means that our f(x) has to be within ε distance of l.0549

We effectively create a boundary that we hold f(x) inside of.0556

Whatever our f(x) ends up being, it has to be somewhere inside of this boundary.0560

If it is outside of the boundary, like this, that is not allowed, because it would fail to be within ε distance of the l.0565

We end up seeing...we have that our l is here, and then the ε's here are just the distance that we end up having for that boundary.0574

We are setting a boundary that is ε distance out from the l.0583

But notice: it has to be true for any ε that we set, so we will be able to create a boundary at whatever size we decide to make,0587

whether it is a giant ε boundary or a really tiny ε boundary.0593

It will always end up working for this next part that we are about to work towards.0596

OK, the second half of the definition (continuing with that): previously, we saw ε as a boundary around the height l.0600

The absolute value of f(x) - l is less than ε means that f(x) has to be, at most (has to be less than), the distance of ε from l.0607

We know that we have to be somewhere contained inside of this boundary in here.0617

OK, now let's move on to looking at δ.0624

What does δ get us? What we did is went and set up for any real ε greater than 0; we had this boundary get set up.0628

So, with that boundary in mind, we now have to go on to say that there exists some real δ greater than 0,0639

such that, for all x where 0 is less than x - c, which is less than δ, we will end up having this |f(x) - l| < ε be true.0646

So, what we are doing is putting a restriction on the allowed x-values.0657

The δ sets some restriction on it: you can't get any farther from c than δ.0663

So, what we are doing is saying, "Remember: the |x - c| is saying the distance between x and c."0669

So, the distance between x and c has to be less than δ.0677

So, if the distance between x and c is less than δ, that means that we have set up, once again, a boundary of how far our x is allowed to roam.0681

And we know that what has to come out of this is: we have to have...0689

if we set x's within that boundary, we will end up having the |f(x) - l| being less than ε.0693

So, if you use x's from inside of this boundary here, we know that we end up having to be mapped into the f(x) that is within that vertical boundary.0702

We are setting up some set of boundaries on what x is that we can have.0713

And we know for sure that we have to have it so that they end up getting mapped inside of this ε.0718

The other thing to notice is that it says that 0 is less than |x - c|, that 0 is strictly less than the absolute value of x - c.0724

It means that x cannot be equal to c, because then we would have an absolute value of 0.0731

We would have x = c; but since it is a limit, we are not concerned with x = c.0735

Remember: we are not concerned with the destination; we are concerned with the journey.0738

So, that means we care about the part with the boundary, but not the actual thing in the middle.0741

That is what that 0 < |x - c| means: that we are guaranteed that we can't have x equal to c,0745

and |x - c| < δ means that we are bounded δ distance, at maximum, from the c in that middle.0753

And everything inside of it ends up getting mapped inside of this boundary here.0761

It will be easier to see what is going on if we erase some of this.0767

What we are doing here is: we can think of it as: some ε gets created around our l that sets up a boundary.0770

We know that you have to be somewhere on this portion vertically.0781

And then, we set some δ radius around our c, so that it ends up mapping only to those vertical locations there.0785

We set ε, and then we have to be able to always set some δ from that ε0797

that will end up mapping us inside of that vertical chunk--that we say that,0801

if we are close enough to this value of c, we will always come out close enough to that value of l.0805

OK, now the really important idea is that we can do this for any ε greater than 0 whatsoever.0813

That is, for any ε greater than 0, there has to exist some δ greater than 0,0818

such that, for x within δ of c, if we are close enough to c, f(x) will end up getting mapped within ε of l.0825

If you are close enough to c, you will end up being close enough to l.0833

We can imagine this process as a never-ending dialogue of ε challenges against δ defenses.0836

Someone says, "Is it possible to stay within this distance of our l?" and we say, "Yes, as long as we are within this distance of our c."0842

And then, they say, "OK, is it possible to stay within this distance of our l, this ε of our l?"0849

We say, "Yes, as long as we are within δ distance of c."0853

So, some ε gets named; for any ε named, there has to be a δ that will cause us to stay within ε away from that l.0856

Otherwise, if we can't do this, then the limit does not exist.0865

But if we can do it, forever and ever and ever, then that means that the limit does exist.0869

If for any ε that gets named, there is always a δ, then that means that the limit does exist.0872

Now, of course, we can't actually have this process of ε challenge and δ defense; we can't do this forever.0877

We can't do this for eternity, so we have to figure out another way to do this.0882

If we wish to truly prove a limit, we have to prove that there will always be some δ greater than 0,0885

given some ε greater than 0--that if some ε gets named, there is always a method0892

that will create the appropriate δ, so that we can get within ε of that limit l.0896

OK, now what we are going to go onto is: we are going to see both of these ideas.0902

Both of these ideas will be explored in the next scene that will be played out by Sherlock Holmes and Dr. Watson.0906

All right, the setting is late afternoon in Sherlock Holmes's apartment.0912

He and Dr. Watson are lounging about.0918

"Let us play a game, Watson!" "All right, Holmes, what type of game?"0921

"A game of limits." "I do not believe I have ever..."0924

"Do not worry: I shall teach it to you. We begin by setting up the game as follows."0927

"One of us (I, in this case) claim the existence of a limit; I name some function f(x)0932

and some value l that the function will approach as x approaches some number c."0938

"Written out, we have the limit as x goes to c of f(x) is equal to l."0944

"All right, that makes sense enough." "Good; after that, the game in earnest begins."0951

"The opposing player (you, in this case) names an ε greater than 0."0955

"It will then be the first player's job to name some δ greater than 0, such that,0959

for all values of x within δ distance of c, except x = c (we don't care about x = c in this game),0964

they will cause f(x) to be within ε of l."0972

"So, if we use one of those x's that is within δ of c, we know that the f(x) that will be produced by that x will have to be within ε of l."0975

"I think I understand." "Let me write it out: I will name some ε greater than 0 that sets up some boundary around l."0983

"So, we have some l - ε to l + ε boundary; we are going to be in that interval vertically."0990

"Then, you name a δ greater than 0 such that, for x contained within c - δ to c + δ,0997

that horizontal bounding, we will end up getting f(x) contained in l - ε to l + ε;1005

that is to say, the boundary that we set around l in the first place."1012

"I will set some boundary vertically, and then you will respond with some horizontal boundary that will put our function inside of that vertical boundary."1016

"Very good, except one important thing: we do not concern ourselves with x = c."1024

"For this game, x is not equal to c; we never actually consider x equaling c, so x is not equal to c, generally."1034

"All right, I understand: you set up a limit; I challenge it with some ε greater than 0, and then you defend with some δ greater than 0."1042

"One question, though: why can't ε equal 0?"1050

"I understand that δ must be greater than 0, because we must have x not equal to c;1054

if δ is greater than 0, then we know that we can't actually be on top of x = c, so x is not equal to c. But why not ε = 0?"1059

"Because this is a game about limits! If you set ε = 0, we would be stuck on l."1067

"We wouldn't have a boundary around it; we would be stuck there."1074

"We need a boundary around l, so that we can talk about approaching it;1077

the entire idea about a limit is to talk about going towards it, approaching it."1082

"It is similar to why we must have x not equal to c; otherwise there is no approaching it, if we are stuck on top of it."1086

"So, we can't be stuck on top of it, either vertically or horizontally; that is why we can't have ε or δ equal to 0."1092

"Makes sense; so what comes next?" "That is it; we stay with the same limit,1098

but the process of ε challenges and δ defenses repeats for as long as we care to play."1103

"If you manage to give an ε that stumps me, you win! You will have shown the limit false."1109

"Let us begin; what will your limit be?" "I choose the following limit to defend: limit as x goes to 3 of x2 + 2 = 11."1115

"All right, I begin by challenging with ε = 1." "I defend with δ = 0.1."1123

"Since x contained within the interval 2.9 to 3.1 (notice: that is a δ of .1 away from our c of c = 3),1130

if x is contained within 2.9 to 3.1, it will give f(x) contained within 10.41 to 11.61."1141

"And that means I am easily within an ε value of 1 from our l of 11."1148

"Thus, my defense stands: using δ = .1, I stay within ε of 11."1153

"OK, how about ε = 0.05." "I will go with δ = 0.005."1159

"Since x contained within 2.995 to 3.005 causes f(x) to be contained within 10.97 to 11.03, I am within ε of 11, so my defense stands again."1167

"What if I tried larger...I set ε = 10!" "Nice try, old chap, but I already showed that δ = 0.1 works for ε = 1."1179

"Therefore, it must work for ε = 10 as well; if you started out small, and then you get bigger,1190

well, I can just leave my δ the same, and it will work just fine for the larger vertical boundary, as well."1195

"Thus, there is no hope for going larger with your ε.1200

Since I have already shown that it works for a small ε, it will automatically work for a larger ε, as well."1204

"Hmm...fine, deal with this: ε = 0.000001." "Elementary, my dear Watson: δ = 0.0000001."1209

"Notice that, if x is contained within 2.9999999 to 3.0000001, we will have f(x) contained within 10.9999994 to 11.0000006."1220

"That is to say, it will be contained within an ε of 11; thus, my defense is still standing."1237

"All right, Holmes; you have me convinced--I believe that, no matter what ε I say, you will be able to defend with an appropriate δ."1244

"But technically, you haven't proven the limit; we can't play this game for eternity."1251

"So, how can you prove that δ will always exist?"1257

"An astute question, Watson; the answer is by showing that there exists some method1261

to create δ from any given ε, and that the method never fails."1266

"I will tell you the method I am using, and then show that it always works."1272

"Of course, you already have a method." "Naturally; here it is: for any ε greater than or equal to 1,1276

I will simply respond with δ = 0.1; as we discussed earlier, δ = 0.1 works for ε = 1,1281

so it will work for any ε greater than or equal to 1."1288

"On the other hand, for ε between 0 and 1 (that is, 0 < ε < 1), I will use δ = ε/10."1291

"Wait; how do we know that δ is greater than 0?"1301

"Well, we know that ε is greater than 0; and since we are creating δ1304

by dividing ε by 10, we know that δ must be greater than 0, as well."1310

"If ε is greater than 0, then we know that dividing something that is greater than 0 doesn't cause it to become 0,1316

and doesn't cause it to become negative; it simply makes it smaller."1322

"So we know that δ must be greater than 0, as well, because we simply divided1325

something that was already greater than 0 by some positive number--in this case, positive 10."1329

"Now, let me prove to you that δ = ε/10 always works."1333

"Notice that, since we are working with the function x2 + 2,1339

the values farthest from c = 3 will produce the values farthest from l = 11, because 32 + 2 = 11."1343

"Thus, we only have to concern ourselves with x = 3 - δ and x = 3 + δ.1352

"If that didn't quite make sense, Watson, what we have here is that, at some horizontal location 3,1356

we know that it puts out the vertical height of 11."1362

"So, as we get farther and farther to either side, we will end up getting farther and farther to 11."1365

"We know that this is a parabola; we are used to working with parabolas; so we see that, the farther we end up getting away from c = 3,1371

the more extreme we are from c = 3, the more extreme we will be from our l = 11."1378

"That is how 32 + 2 would work; as long as we don't go so incredibly far1384

that we end up wrapping the parabola to the other side, we are safe from this; we don't have to worry about it."1388

"So, as long as we can be absolutely sure of that fact, where we don't to -3 or go past 0, we will be fine."1393

"And we know that δ must be less than or equal to .1, because of our earlier restriction on it always being less than or equal to .1."1400

"So, we know that we can't get too far; the most extreme values for our x will end up being using the whole of δ--1407

that is, we only have to concern ourselves with x = 3 - δ and x = 3 + δ."1414

"Now, I want to point out, before we continue, that this logic is specific to the function we are working with, Watson."1420

"It is specific to working with x2 + 2, the fact that, as we get farther and farther away horizontally,1425

we will get farther and farther away vertically, as well; so looking at maximum horizontal distance means maximum vertical distance."1431

"A different function, though, might require different logic; so we have to think specifically1438

about the function that we are working with, if we are going to prove something."1442

"In this case, though, we see that this method will work."1445

"Thus, we can show that our δ = ε/10 works by showing that, if we use the above x1448

(that is x = 3 - δ and x = 3 + δ--if they both work) in our function,1454

for being within ε of l, if we have that they end up coming out like that,1460

we know without a doubt that our δ will always work, because it worked for the most extreme possible values."1467

"We must show that the absolute value of 11 minus f(x)...in this case, if we are plugging in 3 - δ,1474

and our f(x) is x2 + 2, we would have the absolute value of 11, our l, minus...1481

plugging in our function's maximum, most extreme possible value for x, that is (3 - δ)2 + 2 must be less than ε."1486

"Similarly, over here, for our other extreme value for x, x = 3 + δ,1495

if we plug in our f(x), we have the l that we are going to (11) minus the quantity (3 + δ)2 + 2, is less than ε."1501

"Those are the most extreme values there; we are doing the absolute value of l minus f(x) must be less than ε."1511

"That is what we want to show; so to show that this is the case, we will investigate the left sides of the expressions."1519

"Let's work with the 3 - δ part first; we have |11 - (3 - δ)2 + 2|."1524

"Let's begin by expanding 3 minus δ squared; we expand 3 minus δ squared, and now,1531

we have the absolute value of 11 minus (9 - 6δ + δ2); 3 - δ, squared, gets us 9 - 6δ + δ2."1535

"We have minus here, and we have 9 and 2 inside; so minus 9, minus 2, cancels out the 11."1544

"We are left with -6δ + δ2, but there is still this minus sign, so it distributes on,1551

canceling that and making this negative; and we have the absolute value of 6δ minus δ2, when we plug in x = 3 - δ."1560

"Over here, on the other one, we have |11 - (3 + δ)2 + 2|."1569

"We expand 3 + δ, squared; we get 9 + 6δ + δ2;1576

so now we have |11 - (9 + 6δ + δ2) + 2|; once again, we are subtracting by what is inside of that."1581

"So, the 9 and the 2 go together to combine and take out the 11, leaving us with 6δ + δ2."1588

"The minus sign distributes, and we now have -6δ - δ2 if we had plugged in x = 3 + δ."1593

"So, that is what the distance between l and our f(x) is when we plug in our most extreme possible x's, 3 - δ and 3 + δ."1602

"Continuing from this, we can see that, since 0 is less than δ, which is less than or equal to 0.1--1611

remember: we set that δ must be less than or equal to 0.1 at the beginning,1616

because the largest ε that we were allowed to really care about was ε = 1,1619

and we know that δ = 0.1 works there, so we always kept our δ's smaller than 0.1, and δ must be greater than 0."1624

"We can write the above as 6δ - δ2 and 6δ + δ2."1631

"If you are not quite sure about this, since 6δ...well, we know that δ must be greater than 0;1636

thus, 6δ must be positive, and minus δ2...well, δ2 must be smaller than δ,1641

because δ is less than or equal to .1; if we square something that is smaller than 1,1648

it must end up being smaller than had it not been squared."1655

"So, 6δ - δ2...well, 6δ is positive; minus δ2 is negative, but smaller than 6δ."1659

"So, when we take the absolute value of this expression, we are left simply with 6δ - δ2, even after the absolute value."1666

"Over here, we have -6δ - δ2; well, that means the -6δ and the -δ2 are both negative."1673

"So, they combine forces and, after the absolute value is finished with them, they will end up coming out as positive, so we will have 6δ + δ2."1679

"From this, we can create the following inequalities, the right side, in this case,1687

being because δ < 0.1 implies that δ2 < δ, which we talked about very recently."1690

"If we have 6δ - δ2, well, if we had 7 - 5, that would clearly be less than simply 7, if we got rid of the thing subtracting from it."1696

"So, we just get rid of the thing subtracting from it, and we have 6δ - δ2 < 6δ."1706

"That is certainly true; furthermore, since δ2 is less than δ,1712

we have that 6δ + δ2 < 6δ + δ, which is the same thing as 7δ."1716

"So, we now know that 6δ + δ2 is less than 7δ."1723

"At this point, we can use our δ = ε/10 that we originally set; and we have 6δ = 6ε/10."1727

"Remember: that is less than the most extreme value; and 6ε/10 is clearly less than ε."1736

"Furthermore, the other most extreme value was going to be less than 7δ, and we have 7ε/10,1741

which is indeed less than ε; thus, we have shown that plugging in our most extreme values1746

will show that the distance between f(x) and l is always less than ε."1751

"We plug in our most extreme possible value; that gives us our most extreme possible vertical distance,1756

and it still ends up being less than ε when we work the whole thing through."1761

"So, the method of choosing δ always works; we see that this δ = ε/10 will always work; it will never fail."1765

"Very good, Holmes; my only question is how you decided on setting δ equal to ε/10 in the first place."1774

"How did you think, 'ε/10; that is the thing for me'?"1782

"Indeed, that is the most difficult part of proving a limit."1785

"In general, once you have a sense of how the function works, it helps to set it up as if you knew what δ was."1789

"As you work forward, you will find its requirements."1797

"For example, with this limit, I realized that whatever δ was, whatever I ended up choosing for δ,1800

it would end up having to get to the point where we would see that 6δ + δ2 had to be less than ε."1805

"So, if 6δ + δ2 had to be less than ε, I could restrict δ < 0.1,1812

because a smaller δ will never cause any harm, and see that 6δ + δ21817

would be the same thing as saying that it is less than 7δ; and 7δ would certainly work if δ was ε/10."1823

"I could have gone with ε/7, but I figured, 'Why not have a little bit of extra room?'"1830

"And ε divided by 10 just seems so nice and round;1834

so I saw that δ = ε/10 would do fine, at which point I was prepared to work through the proof."1836

"Ah, I am feeling a mite peckish; nothing like a good proof to develop the appetite--what are you in the mood for, Watson?"1842

"Somehow, I find myself craving Greek." "Excellent--good! A brace of gyros it will be!"1848

And they make their exit for the neighborhood gyro shop.1854

All right, with Sherlock Holmes and Watson having helped us see what is going on here,1857

the idea of an ε challenge and a δ defense, and how to prove this stuff in general,1860

by showing that this δ method will always give us a δ that works, we are ready to start working through some examples.1864

Our first example: Below is the graph of f(x) = xsin(1/x).1870

First, we need to determine if the function has a limit, as x goes to 0, and if so, find it and prove it.1876

We can see, right from the beginning, that it looks like it is going to 0.1883

We can see from this nice, convenient graph that it looks like it is going to end up having a limit.1887

It would appear to be the case; however, we notice that we can't simply plug in x = 0.1893

If we did that, we would have 0sin(1/0); we are dividing by 0, so f(0) does not exist.1900

We can't simply go down that method; we can't simply plug in a number and figure out what is going to come out.1910

So, we have to show this by proving it directly; we have to go and see that this is going to always work.1915

We have to figure out a way; so we can see from the graph that, yes, it makes sense that it is going to end up coming out to be 0.1921

But it will be our job to now prove it through a good proof.1926

Let's try to get an understanding of why this thing looks like this.1931

We can see, from the graph, that it looks like this; but let's see what is going on.1934

What we can do is break this down into multiple pieces.1937

We can first see this as sine of...well, let's look, actually, first, at 1/x; what does 1/x graph as?1940

Well, 1/x is going to end up graphing something like this; we are going to go asymptotic as we approach that vertical axis.1951

Then, how does sine of some value graph?1960

Well, sine of some value ends up graphing like this; it has that nice repeating nature.1965

If you watched the last lesson, you probably ended up seeing this in the previous lesson.1975

We talked about just sin(1/x), and how it went crazy and came really, really close.1979

That is the same idea so far; we have that part going on so far.1983

If we plug these two things in--if we plug 1/x in for sine of t, then what we are concerned with,1987

as we get close to 0, as x goes to 0, is that part that is getting close to that.1992

If we end up looking at that, if we look at sin(1/x), we are going to end up getting this part where,1998

as it is far away from 0, it is going to end up going sort of slowly.2008

But as it gets closer, it is going to speed up, and it is going to start to go crazy.2011

And it is going to bounce back and forth between 1 and -1 really, really fast, because it has to go through all of infinity.2015

We shoot off to infinity as we get towards 0; so since it is shooting off towards infinity, 1/x is shooting off towards infinity inside of the sine.2021

So, the sine is now going to have to speed up faster and faster and faster, because it has to go through all of infinity by the time it makes it to 0.2029

We see the same thing from both sides: it shoots up and gets really, really, really, really fast, bouncing up and down.2035

And so, that is the behavior we see.2040

If it was simply sin(1/x), it wouldn't have a limit, as we talked about in the previous lesson,2041

because it is never going to agree on a single location that it is working towards.2046

However, in this case, we also have x showing up; well, if we graph just x, what does just x look like?2051

Well, that looks just like this; so x times something is going to end up expanding it or shrinking it by whatever x's current value is.2059

As x gets closer and closer to 0, it is going to end up squishing it and squishing it and squishing it, closer and closer to that x-axis.2070

Now, notice: sine of anything always produces values that are going to be between -1 and +1,2076

based on how the unit circle works and what we can see from the graph of just sine.2089

So, what we are going to have is either a positive x, effectively, or a negative x.2095

So, if we graph in that -x...here is -x in green...then as we plug in x times sin(1/x), it is going to get...2099

the maximum value that they can have is whatever the x is that it is next to.2114

So, as it gets far away, it ends up bouncing between these things.2117

But as it gets closer and closer and closer, it ends up bouncing faster and faster and faster and faster.2120

But because there is that x, it shrinks it down and crushes it to go down to 0.2125

So, it is not actually going to be defined at x = 0; but it will end up being crushed into this point right here.2130

So, with all of these ideas in mind, we can now say that the limit as x goes to 0 of x times sin(1/x) equals 0.2137

We figured out that, yes, it does have a limit; it makes sense; we understand what is going on2150

in a thing that now makes sense with this graph.2155

And notice: you can also see that the x is going out like this in the graph.2157

And then, -x goes out here; so this nice, computer-generated graph--we end up seeing the same behavior2165

of the sine bouncing infinitely quickly...it gets trapped between +x and -x, because it has to multiply x times sin(1/x).2172

So now, we see that this is what our limit ends up being.2181

And at this point, we can try to work out a proof here.2184

So, if we are going to prove that this is the case, then it must be that, for any ε greater than 0,2187

there exists some δ that will end up making us within that appropriate distance of our vertical ε.2195

For any ε greater than 0, what δ do we choose--what is the appropriate method?2204

Well, notice: if we go back to our x (here is a graph of x, once again; and I will also map the -x here, as well...2208

sorry; that should be dots all the way through); if we have -x and x mapped here, well, what is the biggest thing that x allows?2221

Well, the biggest thing that x is going to allow that sine to get expanded out to is whatever x currently is.2229

For any ε greater than 0, if we want to talk about the maximum vertical size we can allow,2235

then we can allow x to go out as far as that ε, and that will be the maximum allowed vertical size.2238

So, we can simply say that our δ (since that is how far out our x can go from 0) is simply equal to ε.2244

We set δ = ε now, it is going to be our job...what we want to show now is that,2251

for any f(x) - l, it is going to be less than ε if we have our x within δ distance of our c.2260

All right, so let's start working through this.2276

We can swap out what f is and what l is, and we can swap things out, and we can start working and showing that this is true.2277

f(x) - l...what is the left side there? f(x) is this thing right here, x times sin(1/x), minus l...2286

well, l is just 0 in this case, so minus 0...it just closes up, and we have that it is less than ε.2297

So, at this point, we can look at this and say, "Well, x times sin(1/x)...the absolute value of all of that2304

is just going to be the same thing as the absolute value of x, times the absolute value of sin(1/x)."2308

If we strip any possible negative signs after they have multiplied,2313

or strip any possible negative signs before they multiply, it is not going to have an effect on the product.2316

So, we have that this thing right here...2320

Oh, we don't know that it is less than ε; what we want to show is that it comes out to be less than ε.2321

We want to work towards showing that; I'm sorry about that.2326

The absolute value of xsin(1/x) is just the same thing as |x| times |sin(1/x)|.2328

Well, what is the absolute value of sin(1/x)?2338

Well, the absolute value of sin(1/x)...the absolute value of sine of anything at all is going to be less than or equal to 1.2341

You can't get larger than 1 or larger than -1 when you plug something into sine.2355

The absolute value of sin(1/x) is going to still be less than or equal to 1,2360

except for that specific value of x = 0, which would cause the whole thing to break.2365

But we don't have to worry about that, because it is x going to 0; so we have 0 < x - c.2369

So, we don't have to worry about x = 0; so we know that sin(1/x) ≤ 1 will always be true,2375

because we don't have to worry about actually plugging in x = 0.2380

So, we can now use this information up here; and we have that |x| times |sin(1/x)| is less than or equal to...2383

we swap out that fact that sin(1/x) is less than or equal to 1, and we have |x| times 1.2391

So, |x| times |sin(1/x)| is less than or equal to |x| times simply 1.2397

We have that that is just the same thing as saying x; great.2403

Now, at this point, we also know that x - c is less than δ.2407

Let's use yet another color here; so...well, what is our c?2412

Our c is equal to 0 in this case, because that is what we are going towards.2416

So, we have that the absolute value of x - 0...well, we will just leave it as the absolute value of x...has to be less than δ.2420

The absolute value of x is less than δ; we can now swap that in here.2427

And we have that the absolute value of x is less than δ.2431

What did we set our δ to? We know that δ is equal to ε, because that is what we decided to make our method.2437

We have that that is equal to ε; so at this point, we have now shown that the absolute value of x,2442

times the sine of 1/x, gets to being less than or equal to x, which we know is less than δ, which...2450

δ is equal to ε, but at this point we can stack together our signs.2459

The most extreme sign that we ended up having was that direct strictly less than;2462

so at this point, we now see that the absolute value of x times sin(1/x) is, indeed (close that absolute value), less than ε.2466

So, that always ends up being true, if we use this method.2478

We have now completed our proof; this limit does, indeed, work out.2481

We have shown that whatever ε we end up choosing, there will always be an appropriate δ,2485

if we simply set δ = ε it will end up satisfying any and every ε.2490

Thus, we have completed the proof; the limit does exist; the limit is, indeed, 0.2496

Awesome; all right, the second example (and there are only going to be two of these examples,2500

total, because they clearly take a little while to get through):2503

consider the piecewise function f(x) = 1 when x is rational, -1 when x is irrational.2506

Prove that f(x) has no limit anywhere.2512

The very first thing we want to do here, before we try to prove anything, is understand just what f(x) = 1 when x is rational, -1 when x is irrational, means.2515

First, we need to remember what it means for a number to be rational.2524

A number is rational when it can be put as some a/b, where a and b are integers; that is what it means for a number to be rational.2528

A number is irrational when it can't be expressed as a rational.2540

A rational means any decimal number that has a fixed length to it, and also decimals that end up having repeating patterns.2544

But it will actually be enough, with just a fixed length, to understand what is going on here.2551

2.000000001, stop, is a rational number.2555

An irrational number is a decimal number, or something that can be expressed as a decimal number,2562

where the decimals continue going on forever; they never establish a single pattern,2566

and they just constantly are changing forever and ever and ever.2570

For example, π is 3.1415...stuff going on forever; √2, e...we have seen some irrational numbers that are pretty important.2572

But they are also everywhere total; imagine...we had that rational at 2.000000001,2584

but we could also have an irrational that is right next to it, practically, at 2.000000001...2590

and then a random string forever and ever, like 158234671111583...it just keeps changing and changing and changing.2595

So, we can have an irrational number right next to any rational number.2608

We can get as close as we want to any rational number.2612

And we can get as close as we want to any irrational with a rational, by just putting as many decimals as we want, and then stopping at a certain point.2615

So, the rationals and the irrationals are right next to each other, everywhere total.2621

Every single place on the number line has a rational and then an irrational, effectively, infinitely close to it;2627

and then, a rational right next to that and an irrational right next to that...everything is just infinitely mashed together.2632

2 isn't both a rational and an irrational; 2 is simply rational.2638

But right next to it is an irrational; you can get as close to it as you want, with 2.0000000...random pattern.2642

And then, you can get as close as you want to 2.000000...random pattern with a rational, like 2.0000000000000, stop.2648

In matching the random pattern 2.00000000....match the first 50 thousand digits of the random pattern,2658

and then stop, and that is still going to be a rational number, because it is technically an integer divided by an integer.2664

It will be a large integer divided by a large power of 10, but it is still technically an integer over an integer, so it is going to be a rational number.2668

What that means is that rationals and irrationals are right next to each other everywhere.2676

If we tried to graph this, what would f(x) end up looking like?2682

Well, we would end up having some that looked like this, where, let's say, we use blue when one x is rational;2685

so we are going to end up having...everything that ends up being a rational will come out at 1.2693

There are always going to be points, but between any point and the next point over,2700

it is going to end up having a little, tiny hole for the irrationals.2703

And where do the irrationals show up? Well, they show up at -1.2707

So, we end up having the same thing here; down below them are our irrationals.2709

And so, there are a bunch of irrationals; there are infinitely many irrationals down there; there are infinitely many rationals up here.2714

And they both are just right next to each other.2720

You are constantly jumping back and forth and back and forth and back and forth and back and forth, if we were to try to plot this thing out.2722

It is certainly not continuous, because it is two things simultaneously.2728

So, why can't we have a limit? Now let's try to think about why there will be no limit anywhere.2731

Well, consider: if we tried to talk about some location c, then that means that that location c would have to be in either the top or the bottom.2737

And if it ended up being in either the top or the bottom--let's just say the top for argument's sake--2746

then that means that we can put some ε boundary that is small enough to contain just one of these.2750

We make ε equal to a half or something, so that it can just contain one of these height strings, one of these lines.2757

So, since it can only contain one of these lines, no matter what δ we end up choosing,2765

if we choose any δ, then we are going to end up having points that are both on the bottom and on the top.2770

So, if we were in the top, we can make ε only contain the top;2779

but whatever δ size we choose, since the rationals and the irrationals are right next to each other,2783

if we have some rationals, we have to have some irrationals with it, as well.2787

If we have some irrationals, we have to have some rationals with it, as well.2791

If you catch this stuff, it has both types, no matter what, if you take any interval chunk of it.2794

So, since δ will always produce some interval chunk, it has to have both rationals and irrationals in that interval chunk.2800

We grab a chunk, and it is going to have both the top and the bottom,2806

even though we have restricted our ε to only contain either the top or the bottom.2810

And this is the idea of why there can be no limit anywhere: because we can always choose an ε2814

that is going to only take either the top or the bottom line, but no matter what δ we end up choosing,2820

we are going to have to be in both the top and the bottom lines.2825

So, we won't be stuck just inside of that ε it will fail to end up having a limit.2827

If we are going to actually prove this formally, that will be a little bit difficult.2835

But understanding the idea is 90% of the battle.2839

I would much prefer that you understand this idea, and the proof doesn't make sense,2841

than that you really, really work hard through the proof and have a vague understanding of doing the proof,2846

but you don't understand the idea of the limit.2849

At this point (and probably ever), the most important thing, by far, is understanding the general idea,2851

and having a sense of what is going on, rather than being able to do the specific mechanics.2858

The specific mechanics can always come later; understanding the idea is the goal.2862

All right, but let's work through those specific mechanics.2865

OK, proof by contradiction; how do we show this?2868

We will show by assuming it is possible; and then we will see that, if we were to assume that it is possible, crazily impossible things will occur--2871

that if this were true, that we did have a limit, then it won't work--it just doesn't make any sense.2879

And so, we will have contradicted it, and we will know that it must be the case that the limit cannot exist.2885

So, what we do is begin by assuming (there will be a lot of writing for this; I'm sorry)2890

that there exists some limit that as x goes to c; our f(x) is equal to some value of l.2900

Then, for any ε greater than 0, by the definition of how a limit works,2909

we know that there exists some δ greater than 0 such that...2917

and from here on, when I write "such that," I will probably just use s.t. if I have to write that again...2925

such that, for any 0 < |x - c| < δ--that is to say, an x that is less than δ distance2930

away from whatever our c happens to be--we know that it must be the case that the absolute value of f(x) - l,2943

that is, the distance between the f(x) that gets mapped from that x and l, what our limit goes to--has to be less than ε.2951

So, that has to be true; that is what it means for it to have a limit.2959

If there exists a limit, then it must be that for any ε greater than 0, there will always exist a δ greater than 0,2964

such that if x is within δ of c, f(x) must be within ε of l; that is what the formal definition is.2969

Now, let us consider (let me actually change colors, because now we are changing ideas)...2978

that was the setup to this thing: the next thing is: Consider ε = 1/2.2984

Therefore, there must exist, by this thing up here (for any ε greater than 0, there exists a δ greater than 0)...2995

there exists a δ greater than 0 such that it does this thing here.3003

There exists δ greater than 0, such that it follows this idea right here.3010

OK, now, let's continue to talk about this ε equal to 1/2.3016

Notice: by the absolute value of f(x) - l being less than ε, being less than 1/2, it must be either...3022

well, what are the possible values that can come out of f(x)? 1 and -1.3042

It must be either |1 - l| < 1/2, or |-1 - l| < 1/2.3047

The distance that our f(x) is from l must be less than 1/2; so either 1 - l is less than 1/2, or -1 - l is less than 1/2.3058

That is what |f(x) - l| < 1/2 has to come out meaning.3066

But also notice: only one of these can be true.3071

These two ideas right here, |1 - l| < 1/2 and |-1 - l| < 1/2, can't both be true,3085

because if our l is within 1/2 of -1, it can't possibly also reach to being 1/2 of 1.3094

And if our l is within 1/2 of 1, it can't possibly also be within 1/2 of -1.3101

It has to pick one of the two bands; our l can't be 1/2 away from both bands, because 1 and -1 are two apart.3107

If you are within one of them, you can't be near both of them; so it can't be near both of them.3115

Only one of these can be true; we are going to use that fact very soon.3123

Now, we go on to say, "Well, there existed some δ greater than 0 such that it made 0 < |x - c| < δ, which means that |f(x) - l| < ε."3129

So, by our δ, we know that 0 is less than x - c, which is less than δ,3141

and that that ends up having to mean that the absolute value of f(x) - l has to be less than our 1/2, this thing right here.3155

We know that our δ has to end up making our f(x) within 1/2 of l, since we set our ε as 1/2.3166

But 0 < |x - c| < δ--that is to say, x is within δ of c--must contain rational and irrational x.3173

Since we know that δ has to be greater than 0, and our x is within δ of c--3200

it is within some interval of c - δ to c + δ--well, any interval...remember: rationals and irrationals are right on top of each other;3205

then that means our interval, this interval here, the one set up by 0 < |x - c| < δ, must contain both rational and irrational x-values.3224

If it contains both rational and irrational x-values, we know, because it has to be true for any x in this interval,3235

and since we contain both rational and irrational, we now have 1 and -1 popping out of our f(x).3246

Thus, the absolute value of 1 - l is less than 1/2, and the absolute value of -1 - l is less than 1/2 must be true simultaneously,3252

because we know that, within our δ bound, we are going to have both rationals and irrationals.3269

And anything within that δ bound, by the definition (for any ε greater than 0,3275

there exists δ greater than 0, such that, if you are contained within that boundary of δ,3279

it must map within that boundary of ε) that we are contained within that boundary of δ;3284

we are contained within c - δ to c + δ; but there are going to be rationals and irrationals in there.3289

So, that means that our ε boundary must contain both 1 and -1.3293

So, the absolute value of 1 - δ must be less than 1/2, and the absolute of -1 - δ must be less than 1/2.3299

But only one can be true; this and this are mutually exclusive, because you can't be in both bands at once,3304

if the band can only be a total distance of 1 wide around positive 1 or around -1.3312

1/2 down and 1/2 up don't manage to reach each other; they can't reach across, so you can't end up being in both bands at once with that ε of 1/2.3317

So, it says that they must be true simultaneously; but we know that only one can be true.3326

Since only one can be true, and we know that they have to be true simultaneously, those are impossible statements to be true at the same time.3330

We have a contradiction; there is a contradiction in our logic.3336

And because we know everything else in our logic was flawless, except for our assumption that there existed some limit,3340

we know, therefore, that no δ can exist that will satisfy ε equal to 1/2; so no limit can exist.3347

And we have completed our proof.3365

All right, I think that is really cool; it is not for everyone, but I think it is so beautiful.3367

So, I hope you enjoyed it; I hope you get a cool sense of this.3372

And remember: what we talked about here is not something that is necessary for virtually any students--3374

certainly no student currently at a precalculus level.3381

If you are just in the level of this course, you don't need to know this stuff.3383

I am honest; I won't have to use this stuff at all.3386

It is going to maybe, maybe, maybe end up showing up when you take calculus, but probably not even then.3389

It is really something that you only need if you are getting into advanced math.3395

However, I think that this stuff is so cool; and there is no reason that, if you end up liking this stuff...3398

it is like going and checking out art for me; if you end up liking this stuff, you might as well check it out any time that you are able to enjoy it.3402

It is really cool stuff; it gets your brain thinking in all of these totally new ways.3409

And you can end up exploring this stuff for years and years and years.3413

You can make an entire career out of exploring this stuff.3416

If you think this is kind of cool, you can study mathematics in college, go on to become a mathematician, and just study math for the rest of your life.3418

There is a whole bunch of other stuff to explore; math is really, really cool.3425

All right, see you at Educator.com later--goodbye!3428

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