Simplest answer is,
$|sin \left(\frac{1}{x}\right)| \leq 1$
Multiply with $|x|$, then it will be,
=$|x| |sin \left(\frac{1}{x}\right)| \leq |x|$
$= | x sin\left(\frac{1}{x}\right) | \leq|x|$ , ($|a||b| = |ab| $)
as $\lim_{x \to 0} |x| =0$, so $\lim_{x \to 0}$ $|x sin\frac{1}{x}| \leq 0$
Just think for moment is it even possible for an absolute value function to get a value lesser than zero?
Answer is no, so $\lim_{x \to 0}|x sin\frac{1}{x}| =0$
So the answer is option B.
