Question

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

• A. $4$
• B. $3$
• C. $6$
• D. $5$

Answer 1

Answer: (D)

$5$

As $f\left(x\right)$ is differentiable at $x=1$
$5=\underset{h\to 0}{lim}\frac{f(1+h)}{h}$ assuming 0/0 form

Using L-hospital's rule,
$5=\underset{h\to 0}{lim}\frac{f^{\prime }\left(1\right)}{1}$ $\therefore$ $f^{\prime }\left(1\right)=5$