Showing that if $\lim\limits_{x \to a} f'(x)=A$, then $f'(a)$ exists and equals $A$ 
Let $f : [a; b] \to \mathbb{R}$ be continuous on $[a, b]$ and differentiable in $(a, b)$. Show that if $\lim\limits_{x \to a} f'(x)=A$, then $f'(a)$ exists and equals $A$.

I am completely stuck on it. Can somebody help me please? Thanks for your time.

Let $\epsilon>0$. We want to find a $\delta>0$ such that if $0\lt x-a\lt\delta$ then $\left|\dfrac{f(x)-f(a)}{x-a}-A\right|\lt\epsilon$. If $x\gt a$ then MVT tell us that $\dfrac{f(x)-f(a)}{x-a}=f'(c)$ for some $c\in[a,x]$. 
Now use that $\displaystyle\lim_{c\to a^+}f'(c)=A$ to find that $\delta$.
