# Proof Help regarding Limit Differentiation

Suppose $f’(a) = M$ where $M > 0$. Prove that there is a $\delta>0$ such that if $0<|x-a|<\delta$, then $\frac{f(x)-f(a)}{x-a} > M/2$.

I am probably making this problem harder than it is. I would appreciate a push in the right direction. The solution isn't coming to me. Thanks!

-
If the derivative is $M$, then as $x \to a$ the difference quotient approaches $M$, and so must eventually be larger than $M/2$ when $x$ is near enough to $a$. – coffeemath Nov 8 '12 at 15:41
Consider using LaTeX. – glebovg Nov 8 '12 at 15:43
The way it is phrased now, this is not true. Consider $f(x)=0 \forall x \in \mathbb{R}$, then $f'(a)=0 \forall a \in \mathbb{R}$ but the difference quotient does not exist, so certainly it cannot be positive. – gt6989b Nov 8 '12 at 15:46

If $\lim \frac{f(x)-f(a)}{x-a}-\frac{M}{2}$ is positive (it is equal to $\frac{M}{2}$) then the function where you apllied the limit is positive in a reduced neighborhood of $a$. That is $\frac{f(x)-f(a)}{x-a}-\frac{M}{2}>0$ in a reduced neighborhood , as you wanted to prove.