Question

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

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

Answer 1

Answer: (D)

Given, $f(0)=0$ and $\underset{x\to 0}{\lim }\frac{f(x)}{x}=L$ (exist)
$\because f^{\prime }(0)=\underset{x\to 0}{\lim }\frac{f(x)-f(0)}{x-0}$
$=\underset{x\to 0}{\lim }\frac{f(X)}{y}[f(0)=0$ given ]
$\therefore L=f^{\prime }(0)$