Answer
$0$
Work Step by Step
The polar-coordinates are defined as: $x= r \cos \theta , y = r \sin \theta$ and $r^2=x^2+y^2$
$ \lim\limits_{(x,y) \to (0,0) } |f(x,y)|=\lim\limits_{(x,y) \to (0,0) } \dfrac{x^3-xy^2}{x^2+y^2}$
or, $=\lim\limits_{r \to 0} \dfrac{ r^3 \cos^3 \theta- r \cos \theta r^2 \cos^2 \theta}{r^2}$
or, $=\lim\limits_{r \to 0} r \cos \theta ( \cos^2 \theta - \sin^2 \theta )$
or, $=0$
