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For example, consider the proof of the derivative formula for powers of x: lim h → 0 ( x + h ) n − x n h = n x n − 1 . {\displaystyle \lim _{h\to 0}{\frac {(x+h)^{n}-x^{n}}{h}}=nx^{n-1}.} Applying L'Hôpital's rule and finding the derivatives with respect to h yields nx n −1 as expected, but this computation requires the use of the very ...
The book includes the first appearance of L'Hôpital's rule.The rule is believed to be the work of Johann Bernoulli, since l'Hôpital, a nobleman, paid Bernoulli a retainer of 300₣ per year to keep him updated on developments in calculus and to solve problems he had.
Although the rule did not originate with l'Hôpital, it appeared in print for the first time in his 1696 treatise on the infinitesimal calculus, entitled Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes. [3] This book was a first systematic exposition of differential calculus.
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The example below the infobox shows = .However, L'Hopital's rule cannot be applied to this limit, as the proof of the derivative of depends on knowing precisely this limit.
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