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12. limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS12. limits1 Mark
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.
$0$
b
(b) 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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