Chapter 14: Partial Derivatives - Section 14.2 - Limits and Continuity in Higher Dimensions - Exercises 14.2 - Page 796: 22

$0$

Consider $P(x,y)=xy$ Here, we have the point $P(x,y) \to O(0,0)$ This implies that $xy \to 0$ Let us consider $u=xy$ Then $\lim\limits_{u \to 0}\dfrac{1-\cos u}{u}=\dfrac{0}{0}$ This shows that the limit has Indeterminate form thus, we will have to apply L-Hospital's rule Thus, we get $\lim\limits_{u \to 0}\dfrac{\sin u}{(1)}=\sin (0)=0$