Just so an answer is put here rather than in the comments:

As noted in the original question, the desired limit formula is a well-known, or at least straightforward, consequence of Taylor's theorem with Lagrange remainder **in cases where $f\in C^2$**, i.e. if we have some continuity of the 2nd derivative in a neighbourhood of $0$.

As pointed out by Nate Eldredge in the comments: for functions which are merely twice differentiable at the origin, i.e. where the second derivative exists at $0$ but might not be continuous in a neighbourhood of $0$, one still has Taylor's theorem (of order $2$) with *Peano remainder*:
$$
f(x) = f(0) + f'(0)x + \frac{1}{2}f''(0)x^2 + h(x)x^2
$$
where $h(x)\to 0$ as $x\to 0$. This is enough to obtain the desired limit formula.

*Remark.* I took the liberty of contacting the OP and he said:

I am teaching the first real analysis course
for math majors. In class I
proved that if you know the value of a twice
differentiable function at three points you
know the value of the second derivative at
some point ... and I assumed you need continuity
of the second derivative to prove the
limit formula.

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