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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
If $f(x) = \left\{ \begin{array}{l}{x^2}\sin \frac{1}{x},\;\;\;{\rm{when\,\, }}x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\,\,\,{\rm{when\,\,}}\,x = 0\end{array} \right.$, then
  • A
    $f(0 + 0) = 1$
  • B
    $f(0 - 0) = 1$
  • ✓
    $f$ is continuous at $x = 0$
  • D
    None of these
✓

Answer

Correct option: C.
$f$ is continuous at $x = 0$
c
(c) $\mathop {\lim }\limits_{x \to {0^ + }} \,f(x) = {x^2}\sin \frac{1}{x},$ but $ - 1 \le \sin \frac{1}{x} \le 1$ and $x \to 0$

Therefore, $\mathop {\lim }\limits_{x \to {0^ + }} \,f(x) = 0 = \mathop {\lim }\limits_{x \to {0^ - }} \,f(x) = f(0)$

Hence $f(x)$ is continuous at $x = 0.$

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