## Thomas' Calculus 13th Edition

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.