Mathematics 150AL, Fall 2003

Assignment 10 L'Hopital's Rule

Introduction

Certain limits result in the forms $\frac{0}{0}$ or , which are called indeterminate because they do not guarantee that a limit exists, nor do they indicate what the limit is, if one exists. One strategy for dealing with such forms is to rewrite the expression using an algebraic technique.

Examples

. However, we can divide the numerator and denominator by to get .
. However, we can divide the numerator and denominator by $x^{2}$ to get .

Sometimes algebraic techniques are of little use. L'Hopital's Rule can be used when various indeterminate forms arise.

L'Hopital's Rule

Let and be functions that are differentiable on an open interval containing , except possibly at itself. Assume that for all in , except possibly at itself. If the limit of $\frac{f(x)}{g(x)}$ as approaches produces the indeterminate form $\frac{0}{0}$ , then

provided the limit on the right exists (or is infinite). This result also applies if the limit of $\frac{f(x)}{g(x)}$ as approaches produces any one of the indeterminate forms , , , or .

Examples

, which is indeterminate, so we use L'Hopital's Rule: .
, which is indeterminate, so we use L'Hopital's Rule: , which is indeterminate, so we use L'Hopital's Rule again: .

Other indeterminate forms are $0\cdot \infty$ and $\infty -\infty$ . In such cases, you should try to rewrite the limit to fit the form $\frac{0}{0}$ or .

Indeterminate forms such as $1^{\infty },$ $0^{0},$ and $\infty ^{0}$ arise from limits of functions that have variable bases and exponents. In these cases, if the class software can't evaluate the limit, it may be helpful to find the natural logarithm of the limit and use it to find the limit.

Example

Using direct substitution gives the indeterminate form $1^{\infty }$ . Assume and take the natural logarithm of both sides:

, which has the indeterminate form $0\cdot \infty$ so we rewrite the limit as:

, which has the indeterminate form $\frac{0}{0}$ so we may use L'Hopital's Rule to get:

. Thus, we have shown that $\ln y=1$ so the desired limit .

Problems

Fill in the following table and use your result to estimate the limit. Graph the function to support your result.

$x^{7}e^{-x/10}$

For each of the following, illustrate L'Hopital's rule by graphing both $\frac{f(x)}{g(x)}$ and near to see that these ratios have the same limit as $x\rightarrow 0$ . Also calculate the exact value of the limit.
1. $f(x)=e^{x}-1,$ $g(x)=x^{3}+7x$
2. $f(x)=7x\sin x,$ $g(x)=\sec x-1$
For each of the following, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit using L'Hopital's Rule if necessary. (c) Graph the function and verify the result in part (b).
2. $\frac{\sin x}{x}$
3. $\frac{\cos x-1}{x}$
4. $(\tan x)^{\tan 2x}$