Find the $\lim_{(x,y)\to (0,0)} \frac {x^2+y^4}{|x|+|y|} $

Question

I am solving this problem by using polar coordinate and I got the limit is $0$. But using polar coordinate is not enough to show that this limit does exist. I tried the paths along with $x = 0$, $y=0$ and $x=y$ but all got zero. Is there any way to get the limit of this function or proof this function does not have limit on $(0,0)$?

Answer 1

When $|x|,|y|$ small enough, we have $$\frac {x^2+y^4}{|x|+|y|}<\frac {x^2+y^2}{|x|+|y|}<\frac {x^2+y^2+2|x||y|}{|x|+|y|}=|x|+|y|.$$ So the limit is $$\lim_{(x,y)\to (0,0)} \frac {x^2+y^4}{|x|+|y|}=0.$$