Question:
If $f(x)=x \sin \frac{1}{x}, x \neq 0$, then the value of the function at $x=0$, so that the function is continuous at $x=0$, is
(a) 0
(b) −1
(c) 1
(d) indeterminate
Solution:
(a) 0
Given: $f(x)=x \sin \frac{1}{x}, \quad x \neq 0$
Here,
$\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)=\lim _{x \rightarrow 0} x \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)=0 \times \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)=0$
If $f(x)$ is continuous at $x=0$, then $\lim _{x \rightarrow 0} f(x)=f(0)$.
$\Rightarrow f(0)=0$
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