Calculating the second derivative of a function

If I know that a function f is differentiable over all the real numbers, and that 'a' is a real number, how can I show that

$\displaystyle f''\left( a \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h} \right) - 2f\left( a \right) + f\left( {a - h} \right)}}{{{h^2}}}$?

Here I am ssuming that f''(a) exists. The only way I can think of doing it is related to Taylor's Theorem, but we haven't done that on our course yet - I believe the L'Hospital Rule is to be used for solving this problem, but I'm not sure how to go about it.