L’Hôpital’s Rule

L’Hôpital’s rule, in analysis, procedure of differential calculus for evaluating indeterminate forms such as 0/0 and ∞/∞ when they result from an attempt to find a limit. It is named for the French mathematician Guillaume-François-Antoine, marquis de L’Hôpital, who purchased the formula from his teacher the Swiss mathematician Johann Bernoulli. L’Hôpital published the formula in L’Analyse des infiniment petits pour l’intelligence des lignes courbes (1696), the first textbook on differential calculus.

L’Hôpital’s rule states that, when the limit of f(x)/g(x) is indeterminate, under certain conditions it can be obtained by evaluating the limit of the quotient of the derivatives of f and g (i.e., f′(x)/g′(x)). If this result is indeterminate, the procedure can be repeated.