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