L'Hopital's rule

Some limits put you in a standoff. As x approaches a value, the top of a fraction races toward 0 while the bottom also races toward 0 — and 0 over 0 means nothing on its own; the answer could be anything. L'Hopital's rule is a referee for exactly this kind of tie. Instead of comparing the two quantities directly, it compares how fast each one is changing, and that often settles the contest.

The statement: if lim f(x)/g(x) gives the indeterminate form 0/0 or infinity/infinity as x -> a, and f and g are differentiable near a with g'(x) not 0, then lim f(x)/g(x) = lim f'(x)/g'(x), provided that second limit exists (or is plus or minus infinity). You replace numerator and denominator by their derivatives — not the quotient by its derivative, a crucial distinction — and try again. If you land in 0/0 once more, you may apply the rule a second time.

The rule is widely misused. It applies only to the genuine indeterminate forms 0/0 and infinity/infinity; using it on a limit that is already determinate, like 2/3, gives wrong answers. Forms like 0 times infinity, infinity minus infinity, or 1 to the infinity must first be rewritten algebraically into 0/0 or infinity/infinity before the rule applies. And if lim f'/g' fails to exist, that does not mean the original limit fails to exist — the rule is simply silent there.