Limit of multivariable function including absolute value I have some trouble with this limit:
$$\lim_{(x,y)\to (0,0)} \frac{x^3+y^2}{x^2+|y|}.$$
My first attempt to solve it was with polar coordinates but I couldn't find an expression which was independent of $\varphi$. Now I'm trying to solve it with the triangle inequality so I can find an upper limit that squeezes my expression into $0$. My work so far:
$$\left| \frac{x^3+y^2}{x^2+|y|} \right| \leq \frac{|x^3+y^2|}{|x^2+|y||} \leq \frac{|x^3|+|y^2|}{|x^2|+||y||} = \frac{|x^3|+y^2}{x^2+|y|}.$$
But from here I don't know how to continue. 
Thanks in advance! 

By Cauchy–Schwarz
$$|x^3+y^2|=|xx^2+|y||y||\le\sqrt{x^2+y^2}\sqrt{x^4+y^2}\le \sqrt{x^2+y^2}\sqrt{(x^2+|y|)^2}=\sqrt{x^2+y^2}(x^2+|y|)$$
then
$$\left|\frac{x^3+y^2}{x^2+|y|}\right|\le\sqrt{x^2+y^2}\to 0 \implies \frac{x^3+y^2}{x^2+|y|}\to 0$$
