L'Hôpital's Rule

Some limits of quotients can't be found by substituting, because the numerator and the denominator both approach $0$.

In the simplest case the reason is short. If $f(a) = g(a) = 0$, both are differentiable at $a$, and $g'(a) \ne 0$, then for $x \ne a$ near $a$ (where $g(x) \ne 0$, because $\frac{g(x)}{x - a} \to g'(a) \ne 0$)

If $f'$ and $g'$ are also continuous at $a$, then $\frac{f'(a)}{g'(a)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$, which is the rule.

On a free-response answer, show the form before using the rule: write both limits, $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, in limit notation. Writing "$\frac{0}{0}$" as if it were a value isn't enough.