Proving the Second Derivative Using L'Hopital's Rule

[barksdalemc]

Homework Statement

Show that lim(h-->0) [f(x+h)-2f(x) + f(x-h)]/h^2
is equal to f''(x) for any given value of x where the second derivative exists.


I'm supposed to use L'Hopitals rule for this problem. I did and got
[f(x+h)-f(x-h)]/2h

Now I am stuck. I thought about adding and subtracting say f(x) but don't see how that solves to f''(x)

 

[StatusX]

You're almost there. You can rewrite that as the average of the left and right limits for the derivative of f(x), which will be the same if f is differentiable at x.

 

[Hurkyl]

f(x+h)-f(x-h)

is not the derivative of

f(x+h)-2f(x) + f(x-h)


p.s.: for the record, you have to assume f is differentiable in an entire neighborhood of x in order to invoke L'Hôpital's rule.