Question

Check the limit of the function f(x,y) = 3x^2y/(x^2 + y^2) at origin exist or not

Answer 1

Let us use the polar coordinates $\begin{cases} x=r\cos \theta \\ y=r\sin\theta \end{cases}$ , then the limit $(x,y)\to (0,0)$ corresponds to the limit $r\to 0$. We have

$f(r,\theta)=\frac{3r^3\cos^2\theta \sin \theta}{r^2}=3r\cos^2\theta \sin \theta$

We see that the limit $\lim_{r\to 0} f(r,\theta) =0$ exists and independent of $\theta$ (as $3\cos^2\theta \sin \theta$ is a bounded quantity, so $\lim_{r\to 0}r\cdot (\text{bounded quantity})=0$)