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College Calculus: Level I L'Hopital's Rule

Section 5: Application of Derivatives: Lecture 4 | 23:09 min

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Lecture Comments (6)

	0 answers Post by ansam alfaouri on April 3, 2014 I have a question how to find two functions such as that the limit as x approaches infinity f(x)=infinity and as x aproches the infinity g(x)=infinity
	0 answers Post by Constantin Ficiu on November 11, 2013 Great examples and very well explained. Thank you.
	0 answers Post by Stephanie Sergent on June 27, 2012 could you explain by braking it up into smaller steps?
	0 answers Post by Stephanie Sergent on June 27, 2012 example 6 was too complicated.
	1 answer Last reply by: amera abdo Mon Jan 2, 2012 5:04 PM Post by amera abdo on January 2, 2012 What happens to the -1 at the fourth step? how did u get rid of it?

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L'Hopital's Rule

Start by checking to see if you are dealing with an indeterminate type of limit of the form or . If so, proceed with L’Hopital’s Rule.
Remember that it may be necessary to use L’Hopital’s Rule more than once!
If your limit is of another indeterminate type such as , , , , or , then you can rearrange the problem into a L’Hopital’s Rule form.
If your limit is not indeterminate at all, then simply complete the limit computation by ordinary methods!

L'Hopital's Rule

Given lim_{x → π} [cosx/3x], can L'Hopital's Rule be applied?

Identify conditions
f(π) = − 1
g(π) = 3π

It cannot because f(π) ≠ g(π) ≠ 0

Given lim_{x → π} [x/sinx], can L'Hopital's Rule be applied?

Identify conditions
f(0) = 0
g(0) = 0
f′(0) = 1
g′(0) = cos0
= 1

Yes, L'Hopital's Rule can be applied.

Given lim_{x → 0} [(√{8 − x} − 8)/x], can L'Hopital's Rule be applied? If so, what is the limit?

Identify conditions
f(0) = √{8 − 0} − 8
= 2√2 − 8

It cannot be applied because f(0) ≠ (0) ≠ 0

Given lim_{θ→ π} [(tanθ)/(3cosθ+ 3)], can L'Hopital's Rule be applied? If so, what is the limit?

Identify conditions
g′(π) = − 3sinπ
= 0

It cannot be applied because g′(π) = 0

Given lim_{x → ∞} [(x + 4)/(2 − 5x)], can L'Hopital's Rule be applied? If so, what is the limit?

Identify conditions
f(∞) = ∞+ 4
= ∞
g(∞) = 2 − 5(∞)
= ∞
f′(x) = 1
g′(x) = − 5
Apply L'Hopital's Rule

lim_{x → ∞} [(x + 4)/(2 − 5x)] = [1/( − 5)]

Given lim_{y → − 2} [(7y + 14)/(3y² − 12)], can L'Hopital's Rule be applied? If so, what is the limit?

Identify conditions
f( − 2) = 7y + 14
= 7( − 2) + 14
= 0
g( − 2) = 3y² − 12
= 3( − 2)² − 12
= 3(4) − 12
= 0
f′( − 2) = 7
g′( − 2) = 6( − 2)
= − 12

Apply L'Hopital's Rule
lim_{y − 2} [(7y + 14)/(3y² − 12)] = [7/(−12)]

Given lim_{b → ∞} [(b³ + 8b²)/3], can L'Hopital's Rule be applied? If so, what is the limit?

Identify conditions
g(∞) = 3

It cannot be applied because g(∞) ≠ 0 or ∞

Find lim_{x → ∞} [(3^x)/(9x + 2)]

Identify conditions
f(∞) = 3^∞
= ∞
g(∞) = 9x + 2
= ∞
f′(∞) = (ln3)3^∞
g′(x) = 9
Apply L'Hopital's Rule

lim_{x → ∞} [(3^x)/(9x + 2)] = [((ln3)3^∞ )/9]

Find lim_{x → 0} [(6x^²)/(e^x − 1)]

Identify conditions
f(0) = 6x^²
= 6(9)^²
= 0
g(0) = e^x − 1
= e⁰ − 1
= 1 − 1
= 0
f′(0) = 12(0)
= 0
g′(0) = e⁰
= 1

Apply L'Hopital's Rule
lim_{y → − 2} [(6x^²)/(e^x − 1)] = [0/1] = 0

Given lim_{x → ∞} [(3x^1/3)/(e^x(x² + 1))], can L'Hopital's Rule be applied? If so, what is the limit?

Identify conditions
f(∞) = 3(∞)^1/3
= ∞
g(∞) = e^∞(∞² + 1)
= ∞
f′(∞) = x^{− 2/3}
= ∞^{− 2/3}
= 0
g′(∞) = e^x(2x) + e^x(x²)
= e^∞(2(∞)) + e^∞(∞²)
= ∞
Apply L'Hopital's Rule
lim_x∞ [(3x^1/3)/(e^x(x² + 1))] = [0/(∞)]

= 0

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Answer

L'Hopital's Rule

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

Using L'Hopital's Rule

Informal Definition

Lecture Example 1

Lecture Example 2

Lecture Example 3

Lecture Example 4

Additional Example 5

Additional Example 6

Intro 0:00
Using L'Hopital's Rule 0:09

Informal Definition

Lecture Example 1 1:27
Lecture Example 2 4:00
Lecture Example 3 5:40
Lecture Example 4 9:38
Additional Example 5
Additional Example 6

College Calculus 1 Online Course

Section 1: Overview of Functions
	Review of Functions	26:29
	Compositions of Functions	12:29
Section 2: Limits
	Average and Instantaneous Rates of Change	20:59
	Limit Investigations	22:37
	Algebraic Evaluation of Limits	28:19
	Formal Definition of a Limit	23:39
	Continuity and the Intermediate Value Theorem	19:09
Section 3: Derivatives, part 1
	Limit Definition of the Derivative	22:52
	The Power Rule	26:01
	The Product Rule	14:54
	The Quotient Rule	19:17
	Applications of Rates of Change	17:43
	Trigonometric Derivatives	26:58
	The Chain Rule	23:47
	Inverse Trigonometric Functions	27:05
	Equation of a Tangent Line	15:52
Section 4: Derivatives, part 2
	Implicit Differentiation	30:05
	Higher Derivatives	13:16
	Logarithmic and Exponential Function Derivatives	17:42
	Hyperbolic Trigonometric Function Derivatives	14:30
	Related Rates	29:05
	Linear Approximation	23:52
Section 5: Application of Derivatives
	Absolute Minima and Maxima	18:57
	Mean Value Theorem and Rolle's Theorem	20:00
	First Derivative Test, Second Derivative Test	27:11
	L'Hopital's Rule	23:09
	Curve Sketching	40:16
	Applied Optimization	25:37
	Newton's Method	25:13
Section 6: Integrals
	Approximating Areas and Distances	36:50
	Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus	22:02
	Substitution Method for Integration	23:19
Section 7: Application of Integrals, part 1
	Area Between Curves	19:59
	Volume by Method of Disks and Washers	24:22
	Volume by Method of Cylindrical Shells	30:29
	Average Value of a Function	16:31
Section 8: Extra
	Graphs of f, f', f''	23:58
	Slope Fields for Differential Equations	18:32
	Separable Differential Equations	17:04

Related Books

	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 0547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.

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Professor Switkes

L'Hopital's Rule

Slide Duration:

Table of Contents

Section 1: Overview of Functions

Review of Functions

26m 29s

Intro