In my textbook, there is a problem in chapter Expansion of a Function that is
If $f''$ is continuous on some NBD of $c$ prove that,
$\lim_{h\to 0} \frac{f(c+h) - 2f(c) + f(c-h)}{h^2} = f''(c)$
In another chapter Indeterminate Form there is a same looking problem where $f(x)$ is twice differentiable in an NBD of $a$ then the same limit function is given.
My Thinking: These two questions are very same because if a function is differentiable then it is continuous too but not conversely.
I solved it by Taylor 's theorem with $R_2$ and the remainder in Lagrange's form.
But it has been seen that it could be solved by L'Hospital rule, I want to know how.