# Differentiable functions/Mean value theorem/General/L'Hôpital's rule/Section

The following statement is called also the *general mean value theorem*.

Let , and suppose that

are continuous functions which are differentiable on and such that

for all . Then , and there exists some such that

From this version, one can recover the mean value theorem, by taking for the identity.

For the computation of the limit of a function, the following method called *L'Hôpital's rule* helps.

Let denote an open interval, and let denote a point. Suppose that

are continuous functions, which are differentiable on , fulfilling , and with for . Moreover, suppose that the limit

exists. Then also the limit

exists, and it also equals .

Because has no zero in the interval and holds, it follows, because of fact, that is the only zero of . Let denote a sequence in , converging to .

For every there exists, due to
fact,
applied to the interval
or ,
a
(in the interior^{[1]}
of ,)
fulfilling

The sequence converges also to , so that, because of the condition, the right-hand side converges to . Therefore, also the left-hand side converges to , and, because of , this means that converges to .

The polynomials

have both a zero for . It is therefore not immediately clear whether the limit

exists. Applying twice L'Hôpital's rule, we get the existence and

- ↑
The

is the interval without the boundaries.*interior*of a real interval