L'Hôpital's rule provides an efficient technique for using derivatives to evaluate limits of quotients of functions that can otherwise not be determined. It uses derivatives to find limits of functions and different forms can be applied to the two main types of indeterminate forms: / and 0/0.

L'Hôpital's rule is usually written as follows: Let f and g be continuous functions defined on an interval [a, b] and differentiable on (a, b). Suppose that g'(x) 0 for a < x < b and lim_xa+f(x) = lim_xa+g(x) = 0. Then if lim_xa+f'(x)/g'(x) exists, and lim_xa+f(x)/g(x) also exists and the two are equal: lim_xa+f'(x)/g'(x) = lim_xa+f(x)/g(x). This is also true if lim_xa+ is replaced by lim_xb-, or by lim_xc where c is any number in [a, b]. In the case where lim_xc f and g need not be differentiable at c.

L'Hôpital's rule can be applied to many different situations. It can be applied to the two types of indeterminate forms, 0/0 and /. These forms arise when lim_xa+f(x) = 0 = lim_xa+g(x). Then it is said that lim_xa+f(x)/g(x) has the indeterminate form 0/0 because the limit may or may not exist. The same idea applies if lim_xa+ is replaced by lim_xb-, lim_xc, lim_x , or lim_{x -}. L'Hôpital's rule can also be applied to the following situations: limf(x)*g(x), where f(x) approaches infinity and g(x) approaches zero; limf(x)-g(x), where both f(x) and g(x) approach infinity; limf(x)^g(x), where both f(x) and g(x) approach infinity; limf(x)^g(x), where f(x) approaches infinity and g(x) approaches zero; and limf(x)^g(x), where f(x) approaches one and g(x) approaches infinity.

French mathematician Marquis Guillaume F. A. de l'Hôpital is attributed with formulating l'Hôpital's rule. The rule first formally appeared in l'Hôpital's book analyse des infiniment petits pour l'intelligence des lignes courbes published in 1696 and is the first text book concerned with differential calculus. Although the rule first appeared here it should really be called Bernoulli's rule because it first appears in a letter from Johann Bernoulli to l'Hôptal in 1694. L'Hôpital and Bernoulli had made an agreement under which l'Hôpital paid Bernoulli a monthly fee for solutions to particular problems. It was in one of these correspondences that the rule was first formulated. L'Hôpital's first use was in connection to solving: lim_xa [(2a³x-x⁴) -a³(a³x)]/[a - ⁴(ax³)]. The generalized mean value theorem plays an important role in proving l'Hôpital's rule.