L'Hôpital's Rule L'Hôpital's Rule

Introduction

Now that we have experience with derivatives, we will learn how to use the L'Hôpital's Rule to solve a specific type of limit which we otherwise could not solve.

For two functions,

f(x) \text{ and } g(x)
and an input
a
, where

\left[\lim_{x \to a}f(x)=0 \quad \lim_{x \to a}g(x)=0\right]

or

\left[\lim_{x \to a}f(x)=\pm\infty \quad \lim_{x \to a}g(x)=\pm\infty\right]

Then

\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}

Proof for Zeroes

\begin{align*} f(a)&=0 \quad g(a)=0 \\\\ \lim_{x \to a}\frac{f(x)}{g(x)} &= \lim_{x \to a}\frac{f(x) - f(a)}{g(x) - g(a)} \\\\ &= \lim_{x \to a}\frac{f(x) - f(a)}{g(x) - g(a)} * \frac{x-a}{x-a} \\\\ &= \lim_{x \to a}\frac{\frac{f(x) - f(a)}{x-a}}{\frac{g(x) - g(a)}{x-a}} \\\\ &= \frac{f'(a)}{g'(a)} \end{align*}

Proof for Infinities

\begin{align*} \lim_{x \to a} f(x) &= \pm\infty \quad \lim_{x \to a} g(x) = \pm\infty \\\\ \lim_{x \to a} \frac{f(x)}{g(x)} &= \lim_{x \to a} \frac{\frac{1}{g(x)}}{\frac{1}{f(x)}} \\\\ &= \frac{0}{0} \\\\ &= \lim_{x \to a} \frac{\frac{d}{dx}\frac{1}{g(x)}}{\frac{d}{dx}\frac{1}{f(x)}} \\\\ &= \lim_{x \to a} \frac{-\frac{g'(x)}{g(x)^2}}{-\frac{f'(x)}{f(x)^2}} \\\\ &= \lim_{x \to a} \frac{g'(x)}{f'(x)} \cdot \left(\frac{f(x)}{g(x)}\right)^2 \\\\ \lim_{x \to a} \frac{f(x)}{g(x)} &= \lim_{x \to a} \frac{g'(x)}{f'(x)} \cdot \left(\lim_{x \to a} \frac{f(x)}{g(x)}\right)^2 \\\\ \left(\lim_{x \to a} \frac{f(x)}{g(x)}\right)^{-1} &= \lim_{x \to a} \frac{g'(x)}{f'(x)} \\\\ \lim_{x \to a} \frac{f(x)}{g(x)} &= \left(\lim_{x \to a} \frac{g'(x)}{f'(x)}\right)^{-1} \\\\ &= \lim_{x \to a} \frac{f'(x)}{g'(x)} \end{align*}

Examples

Example 1
Evaluate

\lim_{x \to 0} \frac{\sin x}{x}
.
\begin{align*} \lim_{x \to 0} \frac{\sin x}{x} &= \frac{0}{0} \\\\ &= \lim_{x \to 0} \frac{\frac{d}{dx}\sin x}{\frac{d}{dx}x} \\\\ &= \lim_{x \to 0} \frac{\cos x}{1} \\\\ &= \cos(0) \\\\ &= 1 \end{align*}


Example 2
Evaluate
\lim_{x \to \infty} \frac{e^x}{x^2}
\begin{align*} \lim_{x \to \infty} \frac{e^x}{x^2} &= \frac{\infty}{\infty} \\\\ &= \lim_{x \to \infty} \frac{\frac{d}{dx}e^x}{\frac{d}{dx}x^2} \\\\ &= \lim_{x \to \infty} \frac{e^x}{2x} \\\\ &= \frac{\infty}{\infty} \\\\ &= \lim_{x \to \infty} \frac{\frac{d}{dx}e^x}{\frac{d}{dx}2x} \\\\ &= \lim_{x \to \infty} \frac{e^x}{2} \\\\ &= \infty \end{align*}