Prove that $\lim_{(x,y)\to(0,0)}(xy+y^{3})=0$.

Question

Prove that $\lim_{(x,y)\to(0,0)}(xy+y^{3})=0$.

I am trying to determine how to set the $\delta$. Here is my rough work, which isn't much:

$|f(x,y)-0|=|xy+y^{3}|\leq |y||x+y^{2}|$

I am not sure whether I should separate $y$ or separate $xy$ and $y^{3}$ to make it $|xy| + |y^{3}|$. Any ideas how to finish?

Answer 1

Hint

For $\vert x \vert, \vert y \vert \le 1$, you have

$$\vert xy+y^3 \vert \le \vert x \vert \vert y \vert + \vert y \vert^3 \le \vert x \vert \vert y \vert + \vert y \vert\le 2 \vert y \vert$$

Answer 2

Using polar coordinates (not really necessary here, but useful in harder cases): $$|xy + y^3| = |r^2\cos\theta\sin\theta + r^3\sin^3\theta|\le r^2 + r^3,$$ and $r = \|(x,y)\|$ (euclidean norm), so...