f:[a,b]->R is continuous on [a,b] and differentiable on (a,b). Show $\displaystyle \lim_{x \rightarrow\ a}f'(x)=A$, then f'(a) exists and equals A.

The hint we got was to use the definition of f'(a) and the Mean Value theorem, but I"m still having no luck. Can someone please help?