L'Hôpital's rule

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In calculus, l'Hôpital's rule (also called Bernoulli's rule) uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy evaluation of the limit. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital, who published the rule in his book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes (literal translation: Analysis of the Infinitely Small to Understand Curved Lines) (1696), the first textbook on differential calculus.[1] However, it is believed that the rule was discovered by the Swiss mathematician Johann Bernoulli.[2]

The Stolz-Cesàro theorem is a similar result involving limits of sequences, but it uses finite difference operators rather than derivatives.

In its simplest form, l'Hôpital's rule states that for functions f and g:

If \lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0 \, or \pm\infty and \lim_{x\to c}f'(x)/g'(x) exists,

then \lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}.

The differentiation of the numerator and denominator often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be evaluated more easily.


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