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Lecture Comments (7)

	1 answer Last reply by: Micheal Bingham Mon Jul 27, 2015 2:01 PM Post by Scott Beck on April 24, 2014 Shouldn't the pre-set limit rule be (1-cos x)/x rather than what you have?
	1 answer Last reply by: Micheal Bingham Mon Jul 27, 2015 2:01 PM Post by Samiha Bushra on February 18, 2014 I don't understand how sinx/x is equivalent to 1. sin(0) is 0 and if x is 0, shouldn't it be 0 because anything divided by 0 is 0?
	0 answers Post by GUOJI LIU on November 6, 2012 take look the graph f(x)=cos(x), you will find out who when x approaches to 0, the value of function will be 1
	1 answer Last reply by: GUOJI LIU Tue Nov 6, 2012 10:50 AM Post by Kirby Baker on September 27, 2012 how is cos(x) equail to 1

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Special Trigonometric Limits

Reorder terms to resemble special trig limit forms
Useful identity:

Special Trigonometric Limits

Find lim_{x → 0} [1/x] (x − [(x³)/6] + [(x⁵)/120] − [(x⁷)/5040])

lim_{x → 0} [1/x] (x − [(x³)/6] + [(x⁵)/120] − [(x⁷)/5040]) = lim_{x → 0} ([x/x] − [(x³)/6x] + [(x⁵)/120x] − [(x⁷)/5040x])
= 1 − 0 + 0 − 0 = 1
The expression on the right of the original limit just happens to be an approximation of the series representation of sin(x). This representation is accurate to within .0003% for −1 < x < 1.
sin(x) = x − [(x³)/3!] + [(x⁵)/5!] − [(x⁷)/7!] + ... for all x
Reference: the exclamation point is the factorial symbol. Roughly speaking, it equates to one times every integer less than or equal to that number that is still greater than zero. 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

Find lim_{x → 0} [sin(2x)/x]

lim_{x → 0} [sin(2x)/x] = lim_{x → 0} [1/x] (2x − [((2x)³)/6] + [((2x)⁵)/120] − [((2x)⁷)/5040] + ...)
= lim_{x → 0} ([2x/x] − [((2x)³)/6x] + [((2x)⁵)/120x] − [((2x)⁷)/5040x] + ...)
= lim_{x → 0} [2x/x] + 0
= 2 lim_{x → 0} [x/x] = 2 * 1 = 2
In general, lim_{x → 0} [sin(ax)/x] = a

Find lim_{x → 0} [sin(9x)/sin(7x)]

lim_{x → 0} [sin(9x)/sin(7x)] = lim_{x → 0} [sin(9x)/sin(7x)] [x/x]
= lim_{x → 0} [sin(9x)/x] [x/sin(7x)]
= lim_{x → 0} [sin(9x)/x] lim_{x → 0} [x/sin(7x)]
= 9 [1/7] = [9/7]
In general, lim_{x → 0} [sin(ax)/sin(bx)] = [a/b]

[9/7]

Find lim_{x → 0} [1/x] ((1 − [(x²)/2] + [(x⁴)/12] − [(x⁶)/720]) − 1)

lim_{x → 0} [1/x] ((1 − [(x²)/2] + [(x⁴)/12] − [(x⁶)/720]) − 1) = lim_{x → 0} [1/x] (− [(x²)/2] + [(x⁴)/12] − [(x⁶)/720])
= lim_{x → 0} (− [(x²)/2x] + [(x⁴)/12x] − [(x⁶)/720x])
= 0
The bigger exponents in the numerators dominate. In the inner parenthesis, an approximation of the series representation of cos(x) can be seen.
cos(x) = 1 − [(x²)/2!] + [(x⁴)/4!] − ... for all x

= 0

Find lim_{x → 0} [(cos² x − 1)/13x sin x]

lim_{x → 0} [(cos² x − 1)/13x sin x] = [1/13] lim_{x → 0} [(cos² x − 1)/x sin x]
= [1/13] lim_{x → 0} [(−sin² x)/x sin x]
= −[1/13] lim_{x → 0} [sin x/x] = −[1/13]

−[1/13]

Find lim_{x → 0} [sin(2x)/x cos(x)]

lim_{x → 0} [sin(2x)/x cos(x)] = lim_{x → 0} [sin(2x)/x] lim_{x → 0} [1/cos(x)]
= 2 * 1 = 2

Find lim_{x → 0} [sin(2x)/cos(x)sin(3x)]

lim_{x → 0} [sin(2x)/cos(x)sin(3x)] = lim_{x → 0} [2sin(x)cos(x)/cos(x)sin(3x)]
= 2 lim_{x → 0} [sin(x)/sin(3x)]
= [2/3]

= [2/3]

Find lim_{x → 0} [(sin(2x) − 2sin(x))/x sin(x)]

lim_{x → 0} [(sin(2x) − 2sin(x))/x sin(x)] = lim_{x → 0} [(2sin(x)cos(x) − 2sin(x))/x sin(x)]
= lim_{x → 0} [(2cos(x) − 2)/x]
= 2 lim_{x → 0} [(cos(x) − 1)/x]
= 2 * 0 = 0

Find lim_{x → 0} [(5cos(x)sin(x) − 5sin(x))/(x²)]

lim_{x → 0} [(5cos(x)sin(x) − 5sin(x))/(x²)] = lim_{x → 0} [(5sin(x)(cos(x) − 1))/(x²)]
= 5 lim_{x → 0} [sin(x)/x] lim_{x → 0} [(cos(x) − 1)/x]
= 5 * 1 * 0 = 0

Find lim_{x → 0} [tan(5x)/x]

lim_{x → 0} [tan(5x)/x] = lim_{x → 0} [sin(5x)/x cos(5x)]
= lim_{x → 0} [sin(5x)/x] lim_{x → 0} [1/cos(5x)]
= 5 * 1 = 5

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Answer

Special Trigonometric Limits

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

Types of Limits: Special Trig Limits

Pre-set Rules

Special Trig Limits, Example 1

Special Trig Limits, Example 2

Special Trig Limits, Example 3

Special Trig Limits, Example 4: With More Degrees

Special Trig Limits, Example 5

Intro 0:00
Types of Limits: Special Trig Limits 0:07

Pre-set Rules

Special Trig Limits, Example 1 0:58
Special Trig Limits, Example 2 2:50
Special Trig Limits, Example 3 3:55
Special Trig Limits, Example 4: With More Degrees 4:57
Special Trig Limits, Example 5 6:21

Mathematics: AP Calculus AB

Section 1: Functions
	Definitions & Properties of Functions	11:26
	Graphing	13:58
	Inverse Functions	6:47
	Polynomial Functions	5:04
	Trigonometric Functions	6:45
	Inverse Trigonometric Functions	5:58
	Trigonometric Identities	17:42
	Exponential Functions	5:53
	Logarithmic Functions	7:08
	Rational Functions	15:36
	Conic Sections	14:58
Section 2: Limits and Continuity
	Limit Definition & Properties	7:15
	Solving Limits with Algebra	8:01
	Rational Limit Rules	3:16
	One Sided Limits	9:57
	Special Trigonometric Limits	8:28
	Limits & Continuity	10:14
	Limits: Multiple Choice Practice	6:16
Section 3: Derivatives
	Derivative Definition & Properties	4:11
	Basic Rules of Differentiation	7:07
	Power Rule	7:14
	Trigonometric Rules	7:53
	Product Rule	11:11
	Quotient Rule	16:50
	Chain Rule	19:48
	Higher Order Derivatives	15:00
	Derivatives of Exponential Functions	13:14
	Derivatives of Logarithmic Functions	11:30
	Derivatives of Inverse Trigonometric Functions	16:54
	Implicit Differentiation	16:53
	Multiple Choice Practice: Derivatives	11:07
Section 4: Applications of Derivatives
	Tangent & Normal Lines	22:36
	Position Velocity & Acceleration	18:42
	Related Rates	26:22
	Minimum & Maximum	12:22
	Concavity	11:43
	Rolles Theorem	8:28
	Mean Value Theorem	9:39
	Differentials	12:25
	Applications of Derivatives: Multiple Choice Practice	13:21
	Applications of Derivatives: Free Response Practice	10:22
Section 5: Integrals
	Definition of Integrals	1:08
	Integrals of Power Rule	8:50
	Integrals Basic Rules of Integration	9:43
	Trigonometric Rules of Integrals	8:58
	Chain Rule	13:59
	Integrals of Exponential Functions	12:52
	Integrals of Natural Logarithmic Functions	13:00
	Integrals of Inverse Trigonometric Functions	8:29
	Integrals: Multiple Choice Practice	15:37
Section 6: Applications of Integrals
	Fundamental Theorem of Calculus	15:55
	Area Under A Curve	18:34
	Reimann Sums	10:35
	Trapezoid Rule	12:46
	Mean Value Theorem	11:22
	Second Fundamental Theorem of Calculus	4:44
	Area Between Curves	16:39
	Revolving Solids Washer Disk Methods	21:09
	Revolving Solids Cylindrical Shells Method	26:46
	Revolving Solids Known Cross Sections	27:41
	Differential Equations Eulers Method	17:54
	Differential Equations Slope Fields	16:30
	Application of Integrals: Multiple Choice Practice	14:19
	Application of Integrals: Free Response Practice	9:14
Section 7: Sample AP Test
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 1	17:50
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 2	17:32
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 3	22:14
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 4	19:35
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 5	25:43
	AP Calculus AB Practice Test: Section 2: Free Response Part 1	16:50
	AP Calculus AB Practice Test: Section 2: Free Response Part 2	21:36
	AP Calculus AB Practice Test: Section 2: Free Response Part 3	13:39

Related Books

	Cracking the AP Calculus AB Exam, 2015 Edition Author: Princeton Review ISBN: 804124809 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

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John Zhu

Special Trigonometric Limits

Slide Duration:

Table of Contents

Section 1: Functions

Definitions & Properties of Functions

11m 26s

Intro