Please prove the following version of L'Hopital's rule.
Suppose that  f,g : (a,b) \rightarrow \Re
are differentiable with
g(x) and g'(x) never equal to zero.
suppose also that:
lim_{x \rightarrow b-} f(x) =0
lim_{x \rightarrow b-} g(x) =0
lim_{x \rightarrow b-} \frac{f'(x)}{g'(x)} = \infty
then
lim_{x \rightarrow b-} \frac{f(x)}{g(x)} = \infty