Answer
Limit does not exist
Work Step by Step
Consider $f(x,y)=\dfrac{xy}{|xy|}$
Let us consider the approach: $(x,y) \to (0,0)$ along $y=kx; k\ne0$
Then, we get $\lim\limits_{x \to 0} \dfrac{x(kx)}{|x(kx)|}=\lim\limits_{x \to 0}\dfrac{kx^2}{|k|x^2}=\dfrac{k}{|k|}$
This shows that there are multiple limit values when the approach is different and therefore, the limit does not exist for the given function.