L'Hôpital's rule

L'Hôpital's rule (/ˌloʊpiːˈtɑːl/ loh-pee-TAHL) is a mathematical theorem used for evaluating the limit of a quotient of two functions, both 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 ${\displaystyle f}$ and ${\displaystyle g}$, under most circumstances the limit of their quotient can be evaluated as the quotient of the limits: ${\textstyle \lim _{x\to c}f(x)/g(x)={}}$${\textstyle \lim _{x\to c}f(x){\big /}\lim _{x\to c}g(x)}$. This is one of the limit laws. However, if both limits tend to zero (that is, ${\textstyle \lim _{x\to c}f(x)={}}$${\textstyle \lim _{x\to c}g(x)=0}$) or if both tend to infinity, this method cannot be applied because the "indeterminate forms" ⁠${\displaystyle 0/0}$⁠ and ⁠${\displaystyle \infty /\infty }$⁠ are not well defined. L'Hôpital's rule states that in such cases (assuming a non-vanishing derivative in the denominator),$${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}},}$$where ⁠${\displaystyle f'}$⁠ and ⁠${\displaystyle g'}$⁠ are the derivatives of ⁠${\displaystyle f}$⁠ and ⁠${\displaystyle g}$⁠.

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