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$-1$
- DNone of these
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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($A$) $\left(\frac{1}{3}, \frac{1}{\sqrt{3}}\right)$ ($B$) $\left(\frac{1}{4}, \frac{1}{2}\right)$ ($C$) $\left(\frac{1}{3},-\frac{1}{\sqrt{3}}\right)$ ($D$) $\left(\frac{1}{4},-\frac{1}{2}\right)$