# L'Hopital's Rule

## Table of Contents

L’hopital’s rule is a method for evaluating limits of indeterminate form.

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f’(x)}{g’(x)}$$

Where

- \( \small f(x) \) and \( \small g(x) \) are functions,
- \(\small ‘\) denotes the derivative of a function, and
- \( \small c \) is the point at which the limit is being evaluated

## Sources #

## Example #

Calculate the limit of \(\small f(x) = \frac{x^2 - 9}{x - 3}\) as \(\small x\) approaches \(\small 3\).

Since plugging in 3 yields 0/0 which is an indeterminate form, we can apply L’Hopital’s rule, with \( \small f(x) = x^2 - 9\) and \( \small g(x) = x - 3\).

Taking both derivatives, we get \( \small f’(x) = 2x\) and \( \small g’(x) = 1\).

Plugging into the right side of the L’Hopital’s rule, we get that

$$ \lim_{x \to 3} \frac{f(x)}{g(x)} = \lim_{x \to 3} \frac{f’(x)}{g’(x)} = \lim_{x \to 3} \frac{2x}{1} = 6$$