Step in the proof for L’Hopital's rule

Question

This is my first post, so please be easy on me.

I’ve been trying to understand the proof for L’Hopital's rule, which involves breaking down the proof into different cases. I’ve been using this source

I don’t understand this sentence on page 4:

By the souped up MVT, $f/g$ has the same sign as $f′/g′$, so we must have $lim(f/g) = lim(f′/g′)$.

I don’t understand how Cauchy’s mean value theorem (which is what the author calls the souped up MVT) implies that $f/g$ has the same sign as $f′/g′$.

Could anyone explain this to me?

Answer 1

hint

The Cauchy MVT, states that under some conditions,

$$\frac{f(x)-f(a)}{g(x)-g(a)}=\frac{f'(c_x)}{g'(c_x)}$$ When $ x$ goes to $a$, $c_x$ goes also to $ a$.