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