= Solution
<L'Hôpital's rule> applies to a quotient $f(x)/g(x)$ of <differentiable functions> near a limiting <point>, excluding that <point> itself, provided $g'(x)\ne0$ there. If the quotient has an indeterminate form $0/0$ or $\infty/\infty$, and the <limit> of $f'(x)/g'(x)$ exists as a finite or infinite value $L$, then
$$
\boxed{\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}=L.}
$$
The rule also applies to one-sided <limits> and to limits at infinity, under the corresponding hypotheses. Existence of the derivative-ratio <limit> and the indeterminate-form hypotheses are required; an arbitrary quotient cannot be replaced by a quotient of <derivatives>.
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