L'Hôpital's Rule

L'Hôpital's rule states that if $\lim_{x \to a} \frac{f(x)}{g(x)}$ has indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then

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

provided the right-hand limit exists (or is $\pm\infty$).

The rule applies only to those two indeterminate forms. Other indeterminates ($0 \cdot \infty$, $\infty - \infty$, $1^\infty$, $0^0$, $\infty^0$) must first be rewritten into $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form.

The rule may need to be applied repeatedly if the new limit is still indeterminate. It often dramatically simplifies otherwise hard limits, such as $\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1$.