Lebl-Basic-Analysis-I · Problem 4.1.15

The Derivative

Mathematics · lhopital-rule-simple

Problem

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

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

and

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

are differentiable functions whose derivatives

f'

and

g'

are continuous functions. Suppose that at

c \in (a,b)

f(c) = 0

g(c) = 0

g'(x) \neq 0

for all

x \in (a,b)

, and

g(x) \neq 0

whenever

x \neq c

. Note 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)}.