Existence of a limit of a function given that the derivative at the point exists Suppose $f(x)$ is continuous in a neighbourhood of $a$, and $f'(a)$ exists. Does the limit
$$\lim_{x \rightarrow a} \frac{f(x) - f(2a -x)}{2(x-a)}$$
always exist?

$$\begin {align} \lim_{x \mathop \to a} \frac {f(x)-f(2a - x)} {2(x-a)} &= \frac12\lim_{x \mathop \to a} \left(\frac {f(x)-f(a)} {x-a} + \frac {f(a)-f(2a - x)} {x-a}\right)\\&= \frac12\left(\lim_{x \mathop \to a} \frac {f(x)-f(a)} {x-a} + \lim_{x \mathop \to a}\frac {f(a)-f(2a - x)} {x-2a+a}\right)\\&= \frac12\left(\lim_{x \mathop \to a} \frac {f(x)-f(a)} {x-a} + \lim_{u \mathop \to a}\frac {f(a)-f(u)} {-u+a}\right)\end{align}$$
The last equation uses the substitution $u = 2a - x$. Since $f'(a)$ exists you know how to fill out the rest.
