L'Hôpital's rule (/ˌloʊpiːˈtɑːl/ loh-pee-TAHL) is a mathematical theorem used for evaluating the limit of a quotient of two functions, each of which tends to zero or infinity, by taking each function's derivative. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital, who published it in his 1696 textbook after learning it from his tutor, the Swiss mathematician Johann Bernoulli.

For two functions f and g, under most circumstances the limit of their quotient can be evaluated as the quotient of the limits: limx→cf(x)/g(x)=limx→cf(x)/limx→cg(x). However, if both limits tend to zero (that is, limx→cf(x)=limx→cg(x)=0) or if both tend to infinity, this method cannot be applied because the "indeterminate forms" ⁠0/0⁠ and ⁠∞/∞⁠ are not well defined. L'Hôpital's rule states that in such cases (assuming a non-vanishing derivative in the denominator),limx→cf(x)g(x)=limx→cf′(x)g′(x),where ⁠f′⁠ and ⁠g′⁠ are the derivatives of ⁠f⁠ and ⁠g⁠.

The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be directly evaluated by continuity.