Question

Given that $f\left(0\right)=0$ and $\underset{x\to 0}{lim}\frac{f\left(x\right)}{x}$ exists, say $L$. Here $f^{\prime }\left(0\right)$ denotes the derivative of $f$ w.r.t. $x$ at $x=0$. Then $L$ is :

• A. $0$
• B. $2f^{\prime }\left(0\right)-6$
• C. $2f^{\prime }\left(0\right)-5$
• D. $f^{\prime }\left(0\right)$

Answer 1

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

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