# L'Hôpital's rule

## 1 Quick recap

Limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

Examples:

• $$\lim_{x \to 2} x = 2$$
• $$\lim_{x \to 5} (x-1) = 4$$
• $$\lim_{x \to 0} x^2 = 0$$
• $$\lim_{x \to 0} \sin(x) = 0$$

## 2 Fact

Limit of a fraction of two functions equals to the fraction of limits of those two functions.

\begin{equation*} \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)} \end{equation*}

Examples:

• $$\lim_{x \to 3} \frac{x-1}{x+2} = \frac{\lim_{x \to 3}(x-1)}{\lim_{x \to 3}(x+2)} = \frac{2}{5}$$

## 3 Motivation

What if our limit of functions is undefined? In essence, what if our final fraction is $$\frac{0}{0}$$ or $$\frac{\infty}{\infty}$$?

Examples:

• $$\lim_{x \to 0} \frac{\sin(x)}{x} = \frac{\lim_{x \to 0} \sin(x)}{\lim_{x \to 0} x} = \frac{0}{0}$$
• $$\lim_{x \to \infty} \frac{x}{e^x} = \frac{\lim_{x \to \infty} x}{\lim_{x \to \infty} e^x} = \frac{\infty}{\infty}$$

## 4 Definition

Let $$f$$ and $$g$$ be both differentiable (derivatives $$f'$$ and $$g'$$ exist and are defined), if $$\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$$ or $$\pm \infty$$ for all $$x$$ in the domain with $$x \neq a$$, then

\begin{equation*} \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \end{equation*}

## 5 Worked examples

Recall some examples from Motivation

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

This means that as $$x$$ gets infinitely small, $$\sin(x)$$ and $$x$$ tend to have the same value!

## 6 Extra example

Our second example from motivation

\begin{align*} \lim_{x \to \infty} \frac{x}{e^x} & = \lim_{x \to \infty} \frac{(x)'}{(e^x)'} \\ & = \lim_{x \to \infty} \frac{1}{e^x} \\ & = \frac{1}{\lim_{x \to \infty} e^x} = 0 \end{align*}

So as $$x$$ gets infinitely large, $$e^x$$ is getting larger much quicker than $$x$$ does!