Question

If $f^{\prime }\left(c\right)$ exists and non-zero then $\underset{h\to o}{lim}\frac{f(c+h)+f(c-h)-2f\left(c\right)}{h}$ is equal to-

• A. f'(c)
• B. 0
• C. 2f'(c)
• D. None of these

Answer 1

The correct option is B 0

Let, $L=\underset{h\to 0}{lim}\frac{f(c+h)+f(c-h)-2f\left(c\right)}{h}$ $\frac{0}{0}$ form
Thus using L-hospital rule,
$L=\underset{h\to 0}{lim}\frac{f^{\prime }(c+h)+f^{\prime }(c-h)\times -1-0}{1}$
$=\underset{h\to 0}{lim}\frac{f^{\prime }(c+h)-f^{\prime }(c-h)}{1}=f^{\prime }\left(c\right)-f^{\prime }\left(c\right)=0$