Proving L' Hospital's Rule

Question

Referring to the proof L'Hospital's first rule in Bartle and Sherbert, Introduction to Real Analysis, page 189, Fourth Edition.

By the proof can one conclude that if $L$ is finite then $h(x)=L$ for some $x \in (a,c)$, where $h(x)=f'(x)/g'(x)$??

Answer 1

I'm guessing you got this idea from this part of the proof:

Case(a): If $L\in \Bbb R$ and if $\varepsilon > 0$ is given, there exists $c \in (a,b)$ such that $$L- \varepsilon < \dfrac{f'(u)}{g'(u)} < L + \varepsilon \qquad\text{for } u \in (a,c),$$

That statement is just the definition of $$\lim_{x \to a+} \dfrac{f'(x)}{g'(x)} = L$$ which is an assumption for case (a) of the proof. There is nothing in it that requires $h = f'/g'$ to actually equal $L$. It could be always above $L$ or always below $L$. If you choose a smaller $\varepsilon$, it just means that the $c$ promised by the definition will have to be closer to $a$.