Part A: L'Hospital's Rule and Improper Integrals

Course Info

Instructor

Prof. David Jerison

Departments

Mathematics

As Taught In

Fall 2010

Level

Undergraduate

Topics

Mathematics
- Calculus
- Differential Equations

Learning Resource Types

Exams with Solutions

Lecture Notes

Lecture Videos

Problem Sets with Solutions

Simulations

Recitation Videos

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5. Exploring the Infinite

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L’Hospital’s rule describes how we can use the derivative to calculate certain limits. The second half of this part of the course discusses how to calculate the area under a graph when that area is unbounded.

» Session 87: L’Hospital’s Rule
» Session 88: Examples of L’Hospital’s Rule
» Session 89: L’Hospital’s Rule and Rates of Growth
» Session 90: Advanced Examples of L’Hospital’s Rule
» Session 91: Improper Integrals
» Session 92: Integral Comparison
» Session 93: Indefinite Integrals and Singularities

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Overview

Early in the course we discussed limits and things that could go wrong when trying to compute them. For example, we might end up trying to divide zero by zero when calculating a limit. L’Hospital’s rule is the tool we employ when this is unavoidable.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to L’Hospital’s Rule

Clip 2: Elementary Example of L’Hospital’s Rule

Worked Example

sin x/(1 − cos x) as x Approaches 0

Lecture Video and Notes

Video Excerpts

Clip 1: Why L’Hospital’s Rule Works

Clip 2: L’Hospital’s Rule, Version 1

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Overview

In this session we see how L’Hospital’s rule is used and how it is related to quadratic approximation. Finally, we learn what other limits L’Hospital’s rule can be applied to.

Lecture Video and Notes

Video Excerpts

Clip 1: Limit of sin(5x)/sin(2x)

Clip 2: Repeating L’Hospital’s Rule

Clip 3: Comparison With Approximation

Clip 4: Extensions of L’Hospital’s Rule

Worked Example

Limits at Inﬁnity for e^x/x and x/e^x

Recitation Video

L’Hospital Practice

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Overview

If f(x)/g(x) approaches zero as x goes to infinity we know that for large x, g(x) is much larger than f(x). In this session we use L’Hopital’s rule to compare rates of growth of exponential, logarithmic and polynomial functions.

Lecture Video and Notes

Video Excerpts

Clip 1: Rate of Growth of x ln(x)

Clip 2: Rate of Growth of epx

Clip 3: Comparing Growth of ln(x) and x1/3

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Overview

In this session we see the power of L’Hospital’s rule and some pitfalls to avoid. The session concludes with a summary of our findings on rates of growth.

Lecture Video and Notes

Video Excerpts

Clip 1: The Indeterminate Form 00

Clip 2: Limit of sin(x)/(x2)

Worked Example

Limit as x Goes to Infinity of x^(1/x)

Recitation Video

Failure of L’Hospital’s Rule

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Indeterminate Forms

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Lecture Video and Notes

Video Excerpts

Clip 1: L’Hospital’s Rule, Continued

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Overview

Integration allows us to find the area under the curve. Improper integrals let us find this area (if it’s well defined) even when that curve extends to infinity. Knowing whether the area under a graph up to a large, finite value of x is significantly different from the area under the entire graph can help us simplify calculations and avoid approximation errors.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Improper Integrals

Clip 2: Integral of e-kx

Clip 3: Integral of e-x2

Clip 4: Integral of 1/x

Worked Example

Integrating 1/(5x + 2)² from 1 to Inﬁnity

Recitation Video

A Solid With Finite Volume and Inﬁnite Cross Section

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Lecture Video and Notes

Video Excerpts

Clip 1: Integral of 1/(xp)

Recitation Video

Improper Integrals

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Overview

Integral comparison can help determine if an improper integral converges, without calculating the integral.

Lecture Video and Notes

Video Excerpts

Clip 1: Introduction to Comparison

Worked Example

Conﬁrming an Integral Converges

Lecture Video and Notes

Video Excerpts

Clip 1: Functions Without Elementary Anti-Derivatives

Clip 2: Comparison Example

Recitation Video

Integral of xⁿ e^(-x)

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Overview

We can find the area under the curve near a vertical asymptote using a process similar to the one used to find the area under a curve going to infinity. If we forget to do this, we may get a nonsensical value for our integral.