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