Chapter 13 - Section 13.2 - Limits and Continuity in Higher Dimensions - Exercises - Page 692: 58

Yes

Consider $ |\cos \dfrac{1}{y}| \leq 1, |x \cos \dfrac{1}{y}| \leq x$ $\implies 0 \leq |x \cos \dfrac{1}{y}| \leq |x|$ Since, by the Squeeze Theorem $\lim\limits_{(x,y) \to (0,0) }|x \cos \dfrac{1}{y}| =0 $ and $\lim\limits_{(x,y) \to (0,0) } x \cos (\dfrac{1}{y})=0$ Yes, by the Squeeze Theorem the limit for $\lim\limits_{(x,y) \to (0,0) } x \cos (\dfrac{1}{y})=0$ .