Limits Using Polar Coordinates

A method for evaluating limits in two dimensions. For ${\lim }_{(x,y)\to (0,0)}f(x,y)$ Substitute $x=r\cos \theta$, $y=r\sin \theta$ Then, you get ${\lim }_{r\to 0}f(r\cos \theta ,r\sin \theta )$

Example

${\lim }_{(x,y)\to (0,0)}\frac{3x^{2}y}{x^2+y^2}={\lim }_{r\to 0}\frac{3r^2{\cos }^2\theta r\sin \theta }{r^2{\cos }^2\theta +r^2{\sin }^2\theta }={\lim }_{r\to 0}\frac{3r^3\cos \theta \sin \theta }{r^2}=0$