Basic Analysis I — Real Analysis · Problem 4.2.9

The Derivative — Jiri Lebl

Mathematics — Real Analysis · lhopital-rule-mvt

Problem

Prove the following version of L'H\^opital's rule. Suppose

f : (a,b) \to \mathbb{R}

and

g : (a,b) \to \mathbb{R}

are differentiable functions and

c \in (a,b)

. Suppose that

f(c) = 0

g(c) = 0

g'(x) \neq 0

when

x \neq c

, and that the limit of

f'(x)/g'(x)

x

goes to

c

exists. Show that

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.

Compare to Exercise 4.1.15. Note: Before you do anything else, prove that

g(x) \neq 0

when

x \neq c