Answer
Yes ; $ \lim\limits_{(x,y) \to (0,0) } y \sin \dfrac{1}{x} =0$
Work Step by Step
Given : $ |\sin \dfrac{1}{x}| \leq 1, |y \sin \dfrac{1}{x}| \leq y$
Re-arrange as: $0 \leq |y \sin \dfrac{1}{x}| \leq |y|$
Since, by the Squeeze Theorem $\lim\limits_{(x,y) \to (0,0) } |y \sin \dfrac{1}{x}| =0 $
and $\lim\limits_{(x,y) \to (0,0) } y \sin (1/x)=0$
This implies that the limit for $ \lim\limits_{(x,y) \to (0,0) } y \sin \dfrac{1}{x} =0$ by the Squeeze Theorem.
So, our answer is Yes.
