L' Hospital Rule

A rule useful in evaluating indeterminate forms: if both the functions f(x) and g(x) and all their derivatives up to order (n-1) vanish at x=a, but the nth derivatives both do not vanish or both become infinite at x=a, then f⁽ⁿ⁾denoting the nth derivative.

In calculus, l'Hopital'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. Suppose that we want to find the value of , when f (a) = g (a) = 0. One method is to use L'Hopital's Rule, which says: if f(x) and g(x) are differentiable functions and f (a) = g (a) = 0, then , if the limit on the right exists. In other words, we can find the original limit by finding the limit of the ratio of the derivatives of the numerator and denominator functions.

Let lim stand for the limit , , , or , and suppose that lim and lim are both zero or are both. If has a finite value or if the limit is , then

Procedure of differential calculus for evaluating indeterminate forms such as 0/0 and 8/8 when they result from attempting to find a limit. It 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. It is named for the French mathematician Guillaume de L'Hospital (1661- 1704),