Unit content
L'Hôpital's rule
Some limits of quotients remain unresolved after direct substitution. When a quotient approaches the indeterminate form $0/0$ or $\infty/\infty$, derivatives can sometimes reveal the relative rates at which numerator and denominator change.
The rule
Suppose $f$ and $g$ are differentiable near the limiting point, $g'(x)\ne0$ there, and
$$\frac{f(x)}{g(x)}$$
has the form $0/0$ or $\infty/\infty$. Under the usual hypotheses, if the derivative quotient has a limit, then
$$\lim_{x\to a}\frac{f(x)}{g(x)}
\lim_{x\to a}\frac{f'(x)}{g'(x)}.$$
The same principle applies to appropriate one-sided limits and limits at infinity.
Example
Consider
$$\lim_{x\to0}\frac{e^x-1}{x}.$$
Substitution gives $0/0$. Differentiating numerator and denominator gives
$$\lim_{x\to0}\frac{e^x}{1}=1.$$
Therefore the original limit is $1$.
What the rule does not say
L'Hôpital's rule does not mean that
$$\frac{f(x)}{g(x)}=\frac{f'(x)}{g'(x)}.$$
It is a statement about certain limits, not an algebraic identity. It should be applied only after verifying an appropriate indeterminate quotient.
Other indeterminate forms such as $0\cdot\infty$ or $\infty-\infty$ must first be rewritten into a suitable quotient.