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Prove that lim(x->0) (f(x+h)+f(x-h)-2f(x...

Prove that `lim_(x->0) (f(x+h)+f(x-h)-2f(x))/h^2=f''(x)` (without using L' Hospital srule).

Text Solution

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`f''(x)=underset(hrarr0)lim(f'(x+h)-f'(x))/(h)`
`=underset(hrarr0)lim(underset(krarr0)lim[(f(x+h+k)-f(x+h))/(k)-(f(x+k)-f(x))/(k)])/(h)`
Let `k=-h.` Then.
`f''(x)=-underset(hrarr0)lim(f(x)-f(x+h)-f(x-h)+f(x))/(h^(2))`
`=underset(hrarr0)lim(f(x+h)+f(x-h)-2f(x))/(h^(2))`
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