---
tag: 0JWT4PPQ7B
kind: statement
role: theorem
name: L'Hôpital's rule
aliases: [lhopital-rule, bernoulli-rule]
msc: [26A24]
uses: [02SYZAWM2K, 0TXTSEMHR5, 0S7G92KKA7]
sources:
  - url: https://en.wikipedia.org/wiki/L%27Hôpital%27s_rule
    locator: General form
same_as:
  wikidata: Q190557
  wikipedia: L'Hôpital's rule
---

## Statement

Let $I$ be an open interval and let $c \in I$.
Let $f, g$ be real functions differentiable on $I \setminus \{c\}$, with $g'(x) \ne 0$ for all $x \in I \setminus \{c\}$.
Assume that $f(x) \to 0$ and $g(x) \to 0$ as $x \to c$, and that $f'(x)/g'(x) \to L$ as $x \to c$.
Then $f(x)/g(x) \to L$ as $x \to c$.

## Remarks

The proof uses the [[Cauchy mean value theorem|02SYZAWM2K]] on intervals between $c$ and $x$.
Similar forms hold for the [[indeterminate form|0S7G92KKA7]] $\infty/\infty$, for one-sided limits, and for $x \to \pm\infty$.
For example, $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \lim_{x \to 0} \frac{\sin x}{2x} = \frac{1}{2}$.
Mathlib states a form as `deriv.lhopital_zero_nhds`.

## History

Johann Bernoulli found the rule and sent it to Guillaume de l'Hôpital in a letter of 1694.
L'Hôpital published it in his textbook of 1696.

## Explanations

### L1 human

Near $c$, both $f$ and $g$ are close to their tangent lines through $(c, 0)$.
So their ratio is close to the ratio of their slopes.