Question

If $f$ is differentiable function satisfying $f\left(0\right)=0$ and if $g\left(x\right)=\frac{f\left(x\right)}{x}$, then value, that should be assigned to $g\left(0\right)$, so that $g$ is continuous at $0$ is

• A. $1$
• B. $0$
• C. $f\left(0\right)$
• D. $f^{\prime }\left(0\right)$

Answer 1

The correct option is D $f^{\prime }\left(0\right)$

We have, $f\left(0\right)=0,g\left(x\right)=\frac{f\left(x\right)}{x},$

$\underset{x\to 0}{lim}g\left(x\right)=\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}=\underset{x\to 0}{lim}{f}^{\prime }\left(x\right)=f^{\prime }\left(0\right)$

Thus, the required value is $f^{\prime }\left(0\right).$