structural:scholar_edition, structural:summarization
L'Hôpital's rule

L'Hôpital's rule is a general method for evaluating the indeterminate forms 0 / 0 {\displaystyle 0/0} and ∞ / ∞ {\displaystyle \infty /\infty }. This rule states that (under appropriate conditions)

where f ′ {\displaystyle f'} and g ′ {\displaystyle g'} are the derivatives of f {\displaystyle f} and g {\displaystyle g}. (Note that this rule does not apply to expressions ∞ / 0 {\displaystyle \infty /0}, 1 / 0 {\displaystyle 1/0}, and so on, as these expressions are not indeterminate forms.) These derivatives will allow one to perform algebraic simplification and eventually evaluate the limit.

L'Hôpital's rule can also be applied to other indeterminate forms, using first an appropriate algebraic transformation. For example, to evaluate the form 0:

The right-hand side is of the form ∞ / ∞ {\displaystyle \infty /\infty }, so L'Hôpital's rule applies to it. Note that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it is irrelevant how well-behaved f {\displaystyle f} and g {\displaystyle g} may (or may not) be as long as f {\displaystyle f} is asymptotically positive. (the domain of logarithms is the set of all positive real numbers.)

Although L'Hôpital's rule applies to both 0 / 0 {\displaystyle 0/0} and ∞ / ∞ {\displaystyle \infty /\infty }, one of these forms may be more useful than the other in a particular case (because of the possibility of algebraic simplification afterwards). One can change between these forms by transforming f / g {\displaystyle f/g} to (1 / g) / (1 / f) {\displaystyle (1/g)/(1/f)}.