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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).

A

`f^(')(C)`

B

`0`

C

`2f^(')(C)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
`lim_(xrarr0)(f(c+h)+f(c-h)-2f(c))/h`
`=lim_(hrarr0)(f(c+h))-f(c)/h+(f(c-h)-f(c))/h`
`f^(')(c)-lim_(hrarr0)(f(c-h)-f(c))/(-h)=f^(')(c)-f^(')(c)=0`
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