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

MCQ

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

A
$1$
B
$-1$
✓
$0$
D
$2$

✓

Answer

Correct option: C.

$0$

c
(c) 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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