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

Prove that `lim_(h to 0) (f(x+h)+f(x-h)-2f(x))/(h^(2))=f''(x)" (without using L' Hospital's rule)".`

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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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Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as f'(a^-)=lim_(h to 0^(+))(f(a)-f(a-h))/(h) =lim_(hto0^(+))(f(a+h)-f(a))/(h) andf'(a^(+))=lim_(h to 0^+)(f(a+h)-f(a))/(h) =lim_(hto0^(+))(f(a)-f(a+h))/(h) =lim_(hto0^(+))(f(a)-f(x))/(a-x) respectively. Let f be a twice differentiable function. We also know that derivative of a even function is odd function and derivative of an odd function is even function. The statement lim_(hto0)(f(-x)-f(-x-h))/(h)=lim_(hto0)(f(x)-f(x-h))/(-h) implies that for all x"inR ,