If f(x) = l x 1 x , ; ; ; ; ;x 0 ; ; ; ; ; ;0, ; ; ; ; ;x = 0 ., … - Vidyadip

MCQ

If $f(x) = \left\{ \begin{array}{l}x\sin \frac{1}{x},\;\;\;\;\;x \ne 0\\\;\;\;\;\;\;0,\;\;\;\;\;x = 0\end{array} \right.$, then $\mathop {\lim }\limits_{x \to 0} f(x) = $

A
$1$
✓
$0$
C
$-1$
D
None of these

✓

Answer

Correct option: B.

$0$

Here $f(0) = 0$
Since $ - 1 \le \sin \frac{1}{x} \le 1 $
$\Rightarrow - |x| \le x\sin \frac{1}{x} \le |x|$
We know that $\mathop {\lim }\limits_{x \to 0} |x|\, = 0$ and $\mathop {\lim }\limits_{x \to 0} |x|\, = 0$
In this way $\mathop {\lim }\limits_{x \to 0} f(x) = 0.$

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