Question

Given that f(0)= 0 and lim x->0 f(x)/x exists, say L.
Here f'(0) denotes the derivative of f w. r. t. x at
x = 0. Then L is:
(A) 0
(B)2f'(0)-5
(C)2f'(0)-6
(D) f'(0)​

Answer 1

Given,

$\displaystyle\longrightarrow L=\lim_{x\to0}\dfrac{f(x)}{x}$

where $f(x)$ is a function such that $f(0)=0.$

On directly taking $x=0$ we get $L$ as indeterminate form.

$\longrightarrow L=\dfrac{f(0)}{0}=\dfrac{0}{0}$

So we must apply L'hospital's Rule by which we get,

$\displaystyle\longrightarrow L=\lim_{x\to0}\dfrac{\frac{d}{dx}[f(x)]}{\frac{d}{dx}[x]}$

$\displaystyle\longrightarrow L=\lim_{x\to0}\dfrac{f'(x)}{1}$

$\displaystyle\longrightarrow L=\lim_{x\to0}f'(x)$

Now taking $x=0,$

$\displaystyle\longrightarrow\underline{\underline{L=f'(0)}}$

Hence (D) is the answer.