Answer
Yes
Work Step by Step
Given : $2 |xy| - \dfrac{x^2 y^2}{6} \lt 4-4 \cos \sqrt {|xy|} \lt 2 |xy|$
Re-arrange 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 $\lim\limits_{(x,y) \to (0,0) } 2=2$
This implies that the limit for $\lim\limits_{(x,y) \to (0,0) } (\dfrac{ 4-4 \cos \sqrt {|xy|}}{|xy|} )=2$ by the Sandwich Theorem.
So, our answer is Yes.
