If $f(x)$ is differentiable at x, I need to prove that $\lim_{h\to0}\frac{f(x+h)-f(x-h)}{2h}$ exist and is finite.
so if $f(x)$ is differentiable at a $x$, the difference quotient exist for this point, and also $f(x)$ must be continuous at x as well
so that mean that: $\lim_{h\to0^+}\frac{f(x+h)-f(x)}{h} = \lim_{h\to0^-}\frac{f(x+h)-f(x)}{h}$
and that: $\lim_{x\to x_0^+}=\lim_{x\to x_0^-}=f(x_0)$
I know I should probably use arithmetic limit laws to prove this but I can't see how what I figured out could help me. any help with that?
Thanks!
Best Answer
What does it mean that $f$ is differentiable at $x$? It means that $$ \lim_{h\to0} \frac{f(x+h)-f(x)}{h}=f'(x). $$ This is the same (why?) as $$ \lim_{h\to0}\frac{f(x)-f(x-h)}{h}=f'(x). $$ Now add the equations together.