Lesson 17: Interminate forms and L'Hôpital's Rule (worksheet solutions)Matthew Leingang
This document provides solutions to 8 examples of evaluating limits using techniques like L'Hopital's rule and determining the indeterminate form.
The solutions include:
1) Using L'Hopital's rule twice to evaluate a limit of sin^2(x)/xsin(x)+3x^2 as x approaches 0.
2) Identifying a limit of sin(4x)/cos(2x)+1 as a "red herring" since the numerator approaches 0 and denominator approaches a non-zero value.
3) Evaluating a limit without needing L'Hopital's rule by factorizing and simplifying.
4) Applying L'Hopital's rule multiple times and
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
Lesson 17: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document appears to be a lecture on indeterminate forms and L'Hopital's rule from a Calculus I course at New York University. It includes:
1) An introduction to different types of indeterminate forms such as 0/0, infinity/infinity, 0*infinity, etc.
2) Examples of limits that are indeterminate forms and experiments calculating their values.
3) A discussion of dividing limits and exceptions where the limit may not exist even when the numerator approaches a finite number and denominator approaches zero.
4) An outline of the topics to be covered, including L'Hopital's rule, other indeterminate limits, and a summary.
Lesson 17: Interminate forms and L'Hôpital's Rule (worksheet solutions)Matthew Leingang
This document provides solutions to 8 examples of evaluating limits using techniques like L'Hopital's rule and determining the indeterminate form.
The solutions include:
1) Using L'Hopital's rule twice to evaluate a limit of sin^2(x)/xsin(x)+3x^2 as x approaches 0.
2) Identifying a limit of sin(4x)/cos(2x)+1 as a "red herring" since the numerator approaches 0 and denominator approaches a non-zero value.
3) Evaluating a limit without needing L'Hopital's rule by factorizing and simplifying.
4) Applying L'Hopital's rule multiple times and
This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.
Lesson 17: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
This document appears to be a lecture on indeterminate forms and L'Hopital's rule from a Calculus I course at New York University. It includes:
1) An introduction to different types of indeterminate forms such as 0/0, infinity/infinity, 0*infinity, etc.
2) Examples of limits that are indeterminate forms and experiments calculating their values.
3) A discussion of dividing limits and exceptions where the limit may not exist even when the numerator approaches a finite number and denominator approaches zero.
4) An outline of the topics to be covered, including L'Hopital's rule, other indeterminate limits, and a summary.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.
The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.
This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.
This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.
The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.
This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.
The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.
This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.
- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals.
- It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.
The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.
This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.
This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.
This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.
Lesson 17: Indeterminate Forms and L'Hopital's Rule (Section 041 slides)
1. Section 3.7
Indeterminate Forms and L’Hôpital’s
Rule
V63.0121.041, Calculus I
New York University
November 3, 2010
Announcements
Quiz 3 in recitation this week on Sections 2.6, 2.8, 3.1, and 3.2
. . . . . .
2. Announcements
Quiz 3 in recitation this
week on Sections 2.6, 2.8,
3.1, and 3.2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 2 / 34
3. Objectives
Know when a limit is of
indeterminate form:
indeterminate quotients:
0/0, ∞/∞
indeterminate products:
0×∞
indeterminate
differences: ∞ − ∞
indeterminate powers:
00 , ∞0 , and 1∞
Resolve limits in
indeterminate form
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 3 / 34
4. Experiments with funny limits
sin2 x
lim
x→0 x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
5. Experiments with funny limits
sin2 x
lim =0
x→0 x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
6. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim
x→0 sin2 x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
7. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
8. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
.
sin2 x
lim
x→0 sin(x2 )
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
9. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
.
sin2 x
lim =1
x→0 sin(x2 )
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
10. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
.
sin2 x
lim =1
x→0 sin(x2 )
sin 3x
lim
x→0 sin x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
11. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
.
sin2 x
lim =1
x→0 sin(x2 )
sin 3x
lim =3
x→0 sin x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
12. Experiments with funny limits
sin2 x
lim =0
x→0 x
x
lim does not exist
x→0 sin2 x
.
sin2 x
lim =1
x→0 sin(x2 )
sin 3x
lim =3
x→0 sin x
0
All of these are of the form , and since we can get different answers
0
in different cases, we say this form is indeterminate.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 4 / 34
13. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 5 / 34
14. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 5 / 34
15. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
Limit of a product is the product of the limits
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 5 / 34
16. Recall
Recall the limit laws from Chapter 2.
Limit of a sum is the sum of the limits
Limit of a difference is the difference of the limits
Limit of a product is the product of the limits
Limit of a quotient is the quotient of the limits ... whoops! This is
true as long as you don’t try to divide by zero.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 5 / 34
17. More about dividing limits
We know dividing by zero is bad.
Most of the time, if an expression’s numerator approaches a finite
number and denominator approaches zero, the quotient
approaches some kind of infinity. For example:
1 cos x
lim+ = +∞ lim = −∞
x→0 x x→0 − x3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 6 / 34
18. More about dividing limits
We know dividing by zero is bad.
Most of the time, if an expression’s numerator approaches a finite
number and denominator approaches zero, the quotient
approaches some kind of infinity. For example:
1 cos x
lim+ = +∞ lim = −∞
x→0 x x→0 − x3
An exception would be something like
1
lim = lim x csc x.
x→∞ 1 sin x x→∞
x
which does not exist and is not infinite.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 6 / 34
19. More about dividing limits
We know dividing by zero is bad.
Most of the time, if an expression’s numerator approaches a finite
number and denominator approaches zero, the quotient
approaches some kind of infinity. For example:
1 cos x
lim+ = +∞ lim = −∞
x→0 x x→0 − x3
An exception would be something like
1
lim = lim x csc x.
x→∞ 1 sin x x→∞
x
which does not exist and is not infinite.
Even less predictable: numerator and denominator both go to
zero.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 6 / 34
20. Language Note
It depends on what the meaning of the word “is" is
Be careful with the
language here. We are not
saying that the limit in each
0
case “is” , and therefore
0
nonexistent because this
expression is undefined.
0
The limit is of the form ,
0
which means we cannot
evaluate it with our limit
laws.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 7 / 34
21. Indeterminate forms are like Tug Of War
Which side wins depends on which side is stronger.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 8 / 34
22. Outline
L’Hôpital’s Rule
Relative Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 9 / 34
23. The Linear Case
Question
If f and g are lines and f(a) = g(a) = 0, what is
f(x)
lim ?
x→a g(x)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 10 / 34
24. The Linear Case
Question
If f and g are lines and f(a) = g(a) = 0, what is
f(x)
lim ?
x→a g(x)
Solution
The functions f and g can be written in the form
f(x) = m1 (x − a)
g(x) = m2 (x − a)
So
f(x) m
= 1
g(x) m2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 10 / 34
25. The Linear Case, Illustrated
y
. y
. = g(x)
y
. = f(x)
g
. (x)
a
. f
.(x)
. . . x
.
x
.
f(x) f(x) − f(a) (f(x) − f(a))/(x − a) m
= = = 1
g(x) g(x) − g(a) (g(x) − g(a))/(x − a) m2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 11 / 34
26. What then?
But what if the functions aren’t linear?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 12 / 34
27. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear function?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 12 / 34
28. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear function?
What would be the slope of that linear function?
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 12 / 34
29. What then?
But what if the functions aren’t linear?
Can we approximate a function near a point with a linear function?
What would be the slope of that linear function? The derivative!
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 12 / 34
30. Theorem of the Day
Theorem (L’Hopital’s Rule)
Suppose f and g are differentiable functions and g′ (x) ̸= 0 near a
(except possibly at a). Suppose that
lim f(x) = 0 and lim g(x) = 0
x→a x→a
or
lim f(x) = ±∞ and lim g(x) = ±∞
x→a x→a
Then
f(x) f′ (x)
lim = lim ′ ,
x→a g(x) x→a g (x)
if the limit on the right-hand side is finite, ∞, or −∞.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 13 / 34
31. Meet the Mathematician: L'H_pital
wanted to be a military
man, but poor eyesight
forced him into math
did some math on his own
(solved the “brachistocrone
problem”)
paid a stipend to Johann
Bernoulli, who proved this
theorem and named it after
him! Guillaume François Antoine,
Marquis de L’Hôpital
(French, 1661–1704)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 14 / 34
32. Revisiting the previous examples
Example
sin2 x
lim
x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
33. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim
x→0 x x→0 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
34. Revisiting the previous examples
Example . in x → 0
s
.
sin2 x H 2 sin x cos x
lim = lim
x→0 x x→0 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
35. Revisiting the previous examples
Example . in x → 0
s
.
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
36. Revisiting the previous examples
Example . in x → 0
s
.
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x
lim
x→0 sin x2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
37. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example . umerator → 0
n
.
sin2 x
lim
x→0 sin x2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
38. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example . umerator → 0
n
.
sin2 x
lim .
x→0 sin x2
. enominator → 0
d
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
39. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example . umerator → 0
n
.
sin2 x H sin x cos x
2
lim 2.
= lim ( )
x→0 sin x x→0 cos x2 (x)
2
. enominator → 0
d
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
40. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example . umerator → 0
n
.
sin2 x H sin x cos x
2
lim = lim ( )
x→0 sin x2 x→0 cos x2 (x)
2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
41. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example . umerator → 0
n
.
sin2 x H sin x cos x
2
lim = lim ( ) .
x→0 sin x2 x→0 cos x2 (x )
2
. enominator → 0
d
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
42. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example . umerator → 0
n
.
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim ( ) . = lim
x→0 sin x2 x→0 cos x2 (x )
2 x→0 cos x2 − 2x2 sin(x2 )
. enominator → 0
d
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
43. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example . umerator → 1
n
.
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim ( ) = lim
x→0 sin x2 x→0 cos x2 (x)
2 x→0 cos x2 − 2x2 sin(x2 )
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
44. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example . umerator → 1
n
.
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim ( ) = lim .
x→0 sin x2 x→0 cos x2 (x)
2 x→0 cos x2 − 2x2 sin(x2 )
. enominator → 1
d
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
45. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim ( ) = lim =1
x→0 sin x2 x→0 cos x2 (x)
2 x→0 cos x2 − 2x2 sin(x2 )
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
46. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim ( ) = lim =1
x→0 sin x2 x→0 cos x2 (x)
2 x→0 cos x2 − 2x2 sin(x2 )
Example
sin 3x
lim
x→0 sin x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
47. Revisiting the previous examples
Example
sin2 x H 2 sin x cos x
lim = lim =0
x→0 x x→0 1
Example
sin2 x H sin x cos x H
2 cos2 x − sin2 x
lim = lim ( ) = lim =1
x→0 sin x2 x→0 cos x2 (x)
2 x→0 cos x2 − 2x2 sin(x2 )
Example
sin 3x H 3 cos 3x
lim = lim = 3.
x→0 sin x x→0 cos x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 15 / 34
48. Another Example
Example
Find
x
lim
x→0 cos x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 16 / 34
49. Beware of Red Herrings
Example
Find
x
lim
x→0 cos x
Solution
The limit of the denominator is 1, not 0, so L’Hôpital’s rule does not
apply. The limit is 0.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 16 / 34
50. Outline
L’Hôpital’s Rule
Relative Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 17 / 34
51. Limits of Rational Functions revisited
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 3x2 + 7x + 27
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
52. Limits of Rational Functions revisited
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solution
Using L’Hôpital:
5x2 + 3x − 1
lim
x→∞ 3x2 + 7x + 27
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
53. Limits of Rational Functions revisited
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solution
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3
lim = lim
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
54. Limits of Rational Functions revisited
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solution
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3 H 10 5
lim = lim = lim =
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 x→∞ 6 3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
55. Limits of Rational Functions revisited
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solution
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3 H 10 5
lim = lim = lim =
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7 x→∞ 6 3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 18 / 34
56. Limits of Rational Functions revisited II
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 7x + 27
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 19 / 34
57. Limits of Rational Functions revisited II
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 7x + 27
Solution
Using L’Hôpital:
5x2 + 3x − 1
lim
x→∞ 7x + 27
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 19 / 34
58. Limits of Rational Functions revisited II
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 7x + 27
Solution
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3
lim = lim
x→∞ 7x + 27 x→∞ 7
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 19 / 34
59. Limits of Rational Functions revisited II
Example
5x2 + 3x − 1
Find lim if it exists.
x→∞ 7x + 27
Solution
Using L’Hôpital:
5x2 + 3x − 1 H 10x + 3
lim = lim =∞
x→∞ 7x + 27 x→∞ 7
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 19 / 34
60. Limits of Rational Functions revisited III
Example
4x + 7
Find lim if it exists.
x→∞ 3x2 + 7x + 27
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 20 / 34
61. Limits of Rational Functions revisited III
Example
4x + 7
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solution
Using L’Hôpital:
4x + 7
lim
x→∞ 3x2 + 7x + 27
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 20 / 34
62. Limits of Rational Functions revisited III
Example
4x + 7
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solution
Using L’Hôpital:
4x + 7 H 4
lim = lim
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 20 / 34
63. Limits of Rational Functions revisited III
Example
4x + 7
Find lim if it exists.
x→∞ 3x2 + 7x + 27
Solution
Using L’Hôpital:
4x + 7 H 4
lim = lim =0
x→∞ 3x2 + 7x + 27 x→∞ 6x + 7
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 20 / 34
64. Limits of Rational Functions
Fact
Let f(x) and g(x) be polynomials of degree p and q.
f(x)
If p q, then lim =∞
x→∞ g(x)
f(x)
If p q, then lim =0
x→∞ g(x)
f(x)
If p = q, then lim is the ratio of the leading coefficients of f
x→∞ g(x)
and g.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 21 / 34
65. Exponential versus geometric growth
Example
ex
Find lim , if it exists.
x→∞ x2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 22 / 34
66. Exponential versus geometric growth
Example
ex
Find lim , if it exists.
x→∞ x2
Solution
We have
ex H ex H ex
lim = lim = lim = ∞.
x→∞ x2 x→∞ 2x x→∞ 2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 22 / 34
67. Exponential versus geometric growth
Example
ex
Find lim , if it exists.
x→∞ x2
Solution
We have
ex H ex H ex
lim = lim = lim = ∞.
x→∞ x2 x→∞ 2x x→∞ 2
Example
ex
What about lim ?
x→∞ x3
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 22 / 34
68. Exponential versus geometric growth
Example
ex
Find lim , if it exists.
x→∞ x2
Solution
We have
ex H ex H ex
lim = lim = lim = ∞.
x→∞ x2 x→∞ 2x x→∞ 2
Example
ex
What about lim ?
x→∞ x3
Answer
Still ∞. (Why?)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 22 / 34
69. Exponential versus fractional powers
Example
ex
Find lim √ , if it exists.
x→∞ x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 23 / 34
70. Exponential versus fractional powers
Example
ex
Find lim √ , if it exists.
x→∞ x
Solution (without L’Hôpital)
We have for all x 1, x1/2 x1 , so
ex ex
x1/2 x
The right hand side tends to ∞, so the left-hand side must, too.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 23 / 34
71. Exponential versus fractional powers
Example
ex
Find lim √ , if it exists.
x→∞ x
Solution (without L’Hôpital)
We have for all x 1, x1/2 x1 , so
ex ex
x1/2 x
The right hand side tends to ∞, so the left-hand side must, too.
Solution (with L’Hôpital)
ex ex √
lim √ = lim 1 = lim 2 xex = ∞
x→∞ x x→∞ 2 x−1/2 x→∞
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 23 / 34
72. Exponential versus any power
Theorem
Let r be any positive number. Then
ex
lim = ∞.
x→∞ xr
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 24 / 34
73. Exponential versus any power
Theorem
Let r be any positive number. Then
ex
lim = ∞.
x→∞ xr
Proof.
If r is a positive integer, then apply L’Hôpital’s rule r times to the
fraction. You get
ex H H ex
lim = . . . = lim = ∞.
x→∞ xr x→∞ r!
If r is not an integer, let m be the smallest integer greater than r. Then
ex ex
if x 1, xr xm , so r m . The right-hand side tends to ∞ by the
x x
previous step. . . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 24 / 34
74. Any exponential versus any power
Theorem
Let a 1 and r 0. Then
ax
lim = ∞.
x→∞ xr
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 25 / 34
75. Any exponential versus any power
Theorem
Let a 1 and r 0. Then
ax
lim = ∞.
x→∞ xr
Proof.
If r is a positive integer, we have
ax H H (ln a)r ax
lim = . . . = lim = ∞.
x→∞ xr x→∞ r!
If r isn’t an integer, we can compare it as before.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 25 / 34
76. Any exponential versus any power
Theorem
Let a 1 and r 0. Then
ax
lim = ∞.
x→∞ xr
Proof.
If r is a positive integer, we have
ax H H (ln a)r ax
lim = . . . = lim = ∞.
x→∞ xr x→∞ r!
If r isn’t an integer, we can compare it as before.
(1.00000001)x
So even lim = ∞!
x→∞ x100000000
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 25 / 34
77. Logarithmic versus power growth
Theorem
Let r be any positive number. Then
ln x
lim = 0.
x→∞ xr
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 26 / 34
78. Logarithmic versus power growth
Theorem
Let r be any positive number. Then
ln x
lim = 0.
x→∞ xr
Proof.
One application of L’Hôpital’s Rule here suffices:
ln x H 1/x 1
limr
= lim r−1 = lim r = 0.
x→∞ x x→∞ rx x→∞ rx
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 26 / 34
79. Outline
L’Hôpital’s Rule
Relative Rates of Growth
Other Indeterminate Limits
Indeterminate Products
Indeterminate Differences
Indeterminate Powers
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 27 / 34
80. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
81. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hôpital’s Rule:
√
lim x ln x
x→0+
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
82. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hôpital’s Rule:
√ ln x
lim x ln x = lim+ 1 √
x→0+ x→0 / x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
83. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hôpital’s Rule:
√ ln x H x−1
lim x ln x = lim+ 1 √ = lim+ 1
x→0+ x→0 / x x→0 − 2 x−3/2
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
84. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hôpital’s Rule:
√ ln x H x−1
lim x ln x = lim+ 1 √ = lim+ 1
x→0+ x→0 / x x→0 − 2 x−3/2
√
= lim+ −2 x
x→0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
85. Indeterminate products
Example
Find √
lim+ x ln x
x→0
This limit is of the form 0 · (−∞).
Solution
Jury-rig the expression to make an indeterminate quotient. Then apply
L’Hôpital’s Rule:
√ ln x H x−1
lim x ln x = lim+ 1 √ = lim+ 1
x→0+ x→0 / x x→0 − 2 x−3/2
√
= lim+ −2 x = 0
x→0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 28 / 34
86. Indeterminate differences
Example
( )
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
87. Indeterminate differences
Example
( )
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x)
lim+
x→0 x sin(2x)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
88. Indeterminate differences
Example
( )
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim+ = lim+
x→0 x sin(2x) x→0 2x cos(2x) + sin(2x)
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
89. Indeterminate differences
Example
( )
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim+ = lim+
x→0 x sin(2x) x→0 2x cos(2x) + sin(2x)
=∞
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
90. Indeterminate differences
Example
( )
1
lim+ − cot 2x
x→0 x
This limit is of the form ∞ − ∞.
Solution
Again, rig it to make an indeterminate quotient.
sin(2x) − x cos(2x) H cos(2x) + 2x sin(2x)
lim+ = lim+
x→0 x sin(2x) x→0 2x cos(2x) + sin(2x)
=∞
The limit is +∞ becuase the numerator tends to 1 while the
denominator tends to zero but remains positive.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 29 / 34
91. Checking your work
.
tan 2x
lim = 1, so for small x,
x→0 2x
1
tan 2x ≈ 2x. So cot 2x ≈ and
. 2x
1 1 1 1
− cot 2x ≈ − = →∞
x x 2x 2x
as x → 0+ .
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 30 / 34
93. Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Take the logarithm:
( ) ( ) ln(1 − 2x)
ln lim+ (1 − 2x) 1/x
= lim+ ln (1 − 2x)1/x = lim+
x→0 x→0 x→0 x
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 31 / 34
94. Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Take the logarithm:
( ) ( ) ln(1 − 2x)
ln lim+ (1 − 2x) 1/x
= lim+ ln (1 − 2x)1/x = lim+
x→0 x→0 x→0 x
0
This limit is of the form , so we can use L’Hôpital:
0
−2
ln(1 − 2x) H 1−2x
lim+ = lim+ = −2
x→0 x x→0 1
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 31 / 34
95. Indeterminate powers
Example
Find lim+ (1 − 2x)1/x
x→0
Take the logarithm:
( ) ( ) ln(1 − 2x)
ln lim+ (1 − 2x) 1/x
= lim+ ln (1 − 2x)1/x = lim+
x→0 x→0 x→0 x
0
This limit is of the form , so we can use L’Hôpital:
0
−2
ln(1 − 2x) H 1−2x
lim+ = lim+ = −2
x→0 x x→0 1
This is not the answer, it’s the log of the answer! So the answer we
want is e−2 .
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 31 / 34
96. Another indeterminate power limit
Example
lim (3x)4x
x→0
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 32 / 34
97. Another indeterminate power limit
Example
lim (3x)4x
x→0
Solution
ln lim+ (3x)4x = lim+ ln(3x)4x = lim+ 4x ln(3x)
x→0 x→0 x→0
ln(3x) H 3/3x
= lim+ 1 = lim+ −1/4x2
x→0 /4x x→0
= lim+ (−4x) = 0
x→0
So the answer is e0 = 1.
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 32 / 34
98. Summary
Form Method
0
0 L’Hôpital’s rule directly
∞
∞ L’Hôpital’s rule directly
0·∞ jiggle to make 0 or ∞ .
0
∞
∞−∞ combine to make an indeterminate product or quotient
00 take ln to make an indeterminate product
∞0 ditto
1∞ ditto
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 33 / 34
99. Final Thoughts
L’Hôpital’s Rule only works on indeterminate quotients
Luckily, most indeterminate limits can be transformed into
indeterminate quotients
L’Hôpital’s Rule gives wrong answers for non-indeterminate limits!
. . . . . .
V63.0121.041, Calculus I (NYU) Section 3.7 L’Hôpital’s Rule November 3, 2010 34 / 34