Answer
Limit does not exist
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{y^2}{x^2+y^2}$
or, $=\lim\limits_{r \to 0} \cos (\dfrac{ r^2 \sin ^2 \theta}{r^2})$
or, $=\sin^2 \theta $
We know that $\sin^2 \theta $ is not a unique value.
Thus, $ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } \dfrac{y^2}{x^2+y^2}$ =Limit does not exist.
