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

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...
