L'Hôpital's rule

L’Hôpital’s rule allows us to compute limits that arise in the indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$ by passing to the ratio of the derivatives.

Theorem — L'Hôpital

If $\lim_{x\to x_0}\dfrac{f(x)}{g(x)}$ is of the form $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$, and if $\lim_{x\to x_0}\dfrac{f'(x)}{g'(x)}$ exists (finite or infinite), then: $\lim_{x\to x_0}\frac{f(x)}{g(x)} = \lim_{x\to x_0}\frac{f'(x)}{g'(x)}.$

Caution

L’Hôpital applies ONLY when the ratio is in an indeterminate form $0/0$ or $\infty/\infty$. If the limit is in a determinate form, do not use L’Hôpital: you get a wrong result. Moreover, if the limit of the derivatives does not exist, the theorem gives no information (but the original limit might still exist).

Example

$\displaystyle\lim_{x\to 0}\frac{e^x-1}{x}$. Form $\frac{0}{0}$. We apply it: $\dfrac{(e^x)'}{(x)'} = \dfrac{e^x}{1} \to \dfrac{e^0}{1} = 1$.

Example — L'Hôpital applied twice

$\displaystyle\lim_{x\to 0}\frac{e^x-1-x}{x^2}$. Form $\frac{0}{0}$. First application: $\dfrac{e^x-1}{2x}$. Still $\frac{0}{0}$. Second application: $\dfrac{e^x}{2}\to\dfrac{1}{2}$.

Remark

L’Hôpital is very powerful but is not the only method: equivalent functions (seen in the chapter on limits) are often faster. It is best to use L’Hôpital as a “reserve weapon” when equivalent functions are not enough.

Links

Topics: Calculus theorems
Concepts: Derivative · Indeterminate form · L’Hôpital’s theorem
Methods: L’Hôpital limits
Skills: Computing limits
People: Guillaume de l’Hôpital