LHôpitalregeln

L'Hôpital's rule, named after the French mathematician Guillaume de l'Hôpital, is a method used in calculus to evaluate limits of indeterminate forms. Specifically, it applies to limits of the form 0/0 or ∞/∞. The rule states that if the limit of a quotient of two functions, f(x)/g(x), as x approaches a certain value (or infinity) results in an indeterminate form, then the limit is equal to the limit of the quotient of their derivatives, f'(x)/g'(x), provided this latter limit exists or is infinite.

The rule is typically applied when direct substitution of the limit value into the function yields an indeterminate form, meaning the limit cannot be immediately determined. By taking the derivative of both the numerator and the denominator separately, and then evaluating the limit of this new quotient, one can often find the original limit. It is crucial that the derivatives of both the numerator and the denominator exist and that the derivative of the denominator is not zero in the neighborhood of the limit point (except possibly at the point itself). If, after applying the rule once, the limit still results in an indeterminate form, the rule can be applied repeatedly, as long as the conditions for its application are met at each step. This iterative process continues until a determinate form is reached or it's established that the limit does not exist.