L'Hôpital's rule
Some limits arrive in a hopeless-looking shape like 0/0 or infinity/infinity, where plugging in tells you nothing. L'Hôpital's rule offers an escape: replace the numerator and denominator by their slopes — their derivatives — and try the limit again. Often the new ratio is tame and resolves the indeterminate one.
Statement (0/0 case): suppose f and g are differentiable near a (except possibly at a), g'(x) is not 0 near a, and f(x) -> 0 and g(x) -> 0 as x -> a. If lim_{x -> a} f'(x)/g'(x) exists (finite or infinite), then lim_{x -> a} f(x)/g(x) equals that same value. A parallel statement holds for the infinity/infinity form, for one-sided limits, and for x -> infinity. The engine behind the proof is the Cauchy mean value theorem.
Two warnings keep this from being abused. First, it applies only to genuine indeterminate forms; using it on a limit like (x + 1)/(x + 2) at 0, which is plainly 1/2, gives a wrong answer. Second, the rule is a one-way street: if f'/g' fails to have a limit, you may conclude nothing about f/g, which can still converge. It is a tool to try, not a guarantee.
lim_{x -> 0} sin(x)/x is the form 0/0; differentiating top and bottom gives cos(x)/1 -> 1, so the original limit is 1. By contrast lim_{x -> infinity} (x - sin x)/x = 1, yet f'/g' = (1 - cos x)/1 has no limit — L'Hôpital is silent there, and the answer must be found another way.
The rule resolves the first limit but is inconclusive for the second — non-existence of f'/g' is not non-existence of f/g.