Second derivative question

1. The problem statement, all variables and given/known data

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

Homework Helper
2,564
1
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

Staff Emeritus
Science Advisor
Gold Member
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14
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.
 
Last edited:

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