Chapter 13 - Section 13.2 - Limits and Continuity in Higher Dimensions - Exercises - Page 692: 56

Yes

Consider $2 |xy| - \dfrac{x^2 y^2}{6} \lt 4-4 \cos \sqrt {|xy|} \lt 2 |xy|$ It can be rewritten as: $2 -\dfrac{|xy|}{6}\lt \dfrac{ 4-4 \cos \sqrt {|xy|}}{|xy|} \lt 2 $ Since, $\lim\limits_{(x,y) \to (0,0) } (2 -\dfrac{|xy|}{6})=2$ and $\lim\limits_{(x,y) \to (0,0) } 2=2$ Thus, by the Sandwich Theorem: $\lim\limits_{(x,y) \to (0,0) } (\dfrac{ 4-4 \cos \sqrt {|xy|}}{|xy|} )=2$ .