1. Tardigrade
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  3. Mathematics
  4. Given that f(0)= 0 and displaystyle limx → 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 :

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Q. Given that $f(0)= 0$ and $\displaystyle\lim_{x \to 0} \frac{f(x)}{x}$ exists, say L. Here $f'(0)$ denotes the derivative of f w. r. t. x at $x = 0$. Then L is :

UPSEEUPSEE 2017

Solution:

Given, $f(0)=0$ and $\displaystyle\lim _{x \rightarrow 0} \frac{f(x)}{x}=L$ (exist)
$ \because\, f'(0) =\displaystyle\lim _{x \rightarrow 0} \frac{f(x)-f(0)}{x-0} $
$=\displaystyle\lim _{x \rightarrow 0} \frac{f(X)}{y} [f(0)=0 $ given ]
$\therefore \, L=f'(0)$