Please prove the following version of L'Hopital's rule.

Suppose that $\displaystyle f,g : (a,b) \rightarrow \Re $

are differentiable with

$\displaystyle g(x) $ and $\displaystyle g'(x)$ never equal to zero.

suppose also that:

$\displaystyle lim_{x \rightarrow b-} f(x) =0$

$\displaystyle lim_{x \rightarrow b-} g(x) =0$

$\displaystyle lim_{x \rightarrow b-} \frac{f'(x)}{g'(x)} = \infty$

then

$\displaystyle lim_{x \rightarrow b-} \frac{f(x)}{g(x)} = \infty$