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STD 12 - 5. continuity and differentiation — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 5. continuity and differentiation4 Marks
Question
If $f(x) = \left\{ \begin{array}{l}x\sin \frac{1}{x},\,\,x \ne 0\\\,\,\,\,\,\,\,\,\,\,\,\,k,\,\,x = 0\end{array} \right.$ is continuous at $x = 0$, then the value of $k$ is
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Answer

If function $f(x)$ is continuous at $x = 0,$ then$f(0) = \mathop {\lim }\limits_{x \to 0} \,f(x)$
Given $f(0) = k$; $f(0) = k = \mathop {\lim }\limits_{x \to 0} \,x\,\left( {\sin \frac{1}{x}} \right)$
$f(0) = k = 0,{\rm{ }}\left( { - 1 \le \sin \frac{1}{x} \le 1} \right)$;
$\therefore k = 0$.

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