Question

Suppose $f\left(x\right)$ is differentiable $x=1$ and $\underset{h\to 0}{lim}\frac{1}{h}f(1+h)=5$ then $f^{\prime }\left(1\right)$ equals

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is C $5$

$f^{\prime }\left(1\right)=\underset{h\to 0}{lim}\frac{f(1+h)-f\left(1\right)}{h}$; As function is differentiable so it is continuous as it is given that

$\underset{h\to 0}{lim}\frac{f(1+h)}{h}=5$ and hence $f\left(1\right)=0$

$f^{\prime }\left(1\right)=\underset{h\to 0}{lim}\frac{f(1+h)-f\left(1\right)}{h}=5$