Differentiability in relation to strange limit Please do not give a specific answer, just some guidance is what I'm looking for.
This is a problem from an analysis module I am currently taking that has left me scratching my head:
"Suppose that $g:I \to \mathbb{R}$ and that $g$ is differentiable at the point $x=x_0$ does the following limit exist? $\lim_{h \to 0} \frac{g(x_0+h)-g(x_0-h)}{2h}$
Edit: From testing of a few well-defined functions this appears to be some alternative way of expressing a derivative.

The derivate is defined as the limit of the difference quotient. This limit has a geometric interpretation (try doing a sketch for different values getting closer to the limit). ¿Is there any interpretation to your limit?.
If the derivate could be interpeted as the instantaneous velocity, using your interpetation for this new limit, ¿can we make a physical interpretation?.
Does it matter here the value $h(x_0)$?
