Answer
Limit does not exist
Work Step by Step
Consider our approach : $(x,y) \to (0,0)$ along $y=mx; m\ne0$
Then, we get $\lim\limits_{x \to 0} \dfrac{x(mx)}{|x(mx)|}=\lim\limits_{x \to 0}\dfrac{mx^2}{|m|x^2}$
or, $\lim\limits_{x \to 0}\dfrac{mx^2}{|m|x^2}=\dfrac{m}{|m|}$
This shows that there are multiple limit values when the approach is different, so, the limit does not exist at the point (0,0) for the function $f(x,y)=\dfrac{xy}{|xy|}$.
