L'Hôpital's rule ( 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 {\displaystyle f} and g {\displaystyle g} , under most circumstances the limit of their quotient can be evaluated as the quotient of the limits: lim x → c f ( x ) / g ( x ) =
{\textstyle \lim _{x\to c}f(x)/g(x)={}}
lim x → c f ( x ) / lim x → c g ( x ) {\textstyle \lim _{x\to c}f(x){\big /}\lim _{x\to c}g(x)} . However, if both limits tend to zero (that is, lim x → c f ( x ) =
{\textstyle \lim _{x\to c}f(x)={}}
lim x → c g ( x ) = 0 {\textstyle \lim _{x\to c}g(x)=0} ) or if both tend to infinity, this method cannot be applied because the "indeterminate forms" 0 / 0 {\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), lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) , {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}},} where f ′ {\displaystyle f'} and g ′ {\displaystyle g'} are the derivatives of f {\displaystyle f} and g {\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.
History Johann Bernoulli was the original discoverer of this result for the indeterminate form 0 / 0 {\displaystyle 0/0} . He sent it to Guillaume de l'Hôpital (also written l'Hospital) in a letter dated July 22, 1694. Previously, during his time in Paris, Bernoulli signed a contract with l'Hôpital, in which he agreed to teach the new calculus to l'Hôpital, to inform him of his own mathematical discoveries, which l'Hôpital might use as he pleased, and to refrain from sharing his notes with others. In return, l'Hôpital paid Bernoulli an annual allowance, which dramatically improved Bernoulli's financial situation. L'Hôpital published this rule in his 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes (lit. 'Analysis of the Infinitely Small for the Understanding of Curved Lines'). Its original presentation was geometric and L'Hôpital showcased many of the same examples used by Bernoulli. As the first textbook on the differential calculus to be printed, it dominated much of the eighteenth century and helped popularize new mathematics across Europe. The original result was subsequently extended; these are now collectively referred to as l'Hôpital's rules.
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