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L'Hôpital's Rule ○◂｜Definition｜1st｜20260604132713-00-⌔

L’Hôpital’s rule - Wikipedia

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, 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: ${\lim }_{x\to c}f(x)/g(x)=$ ${\lim }_{x\to c}f(x)/{\lim }_{x\to c}g(x)$. However, if both limits tend to zero (that is, ${\lim }_{x\to c}f(x)=$ ${\lim }_{x\to c}g(x)=0$) or if both tend to infinity, this method cannot be applied because the “indeterminate forms” ⁠ $0/0$ ⁠ and ⁠ $\infty /\infty$ ⁠ are not well defined. L’Hôpital’s rule states that in such cases (assuming a non-vanishing derivative in the denominator),

$\underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\underset{x\to c}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)},$

where ⁠ $f^{\prime }$ ⁠ and ⁠ $g^{\prime }$ ⁠ 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.

Printed 2026-06-28.

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