Answer
$0$
Work Step by Step
Consider $P(x,y)=xy$ and the point $P(x,y) \to O(0,0)$
This means that $xy \to 0$
Plug in: $u=xy$
Now, $\lim\limits_{u \to 0}\dfrac{1-\cos u}{u}=\dfrac{0}{0}$, which shows the limit of Indeterminate form; thus, we will apply L-Hospital's rule:
We get $\lim\limits_{u \to 0}\dfrac{\sin u}{(1)}=\sin (0)=0$
