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

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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

The statement

follows from fact. We consider the auxiliary function

We have

Therefore, , and fact yields the existence of some with

Rearranging proves the claim.

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

L’Hospital (1661-1704)

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

  1. The interior of a real interval

    is the interval without the boundaries.