I am studying for a test, and I can not for the life of me figure out why the first example is correct and the second one is not.

$\displaystyle \lim_{(x,y)\rightarrow(0,0)}\frac{xy}{\sqrt{x^{2}+ y^{2}}} = 0 $

since

$\displaystyle 0\leq|{\frac{xy}{\sqrt{x^{2}+y^{2}}}}|\leq|{x}| $

but

$\displaystyle 0\leq|{\frac{2x^{2}y}{x^{4}+y^{2}}}|\leq2|{y}| $

does not work since

$\displaystyle \lim_{(x,y)\rightarrow(0,0)}{\frac{2x^{2}y}{x^{4}+ y^{2}}}\text{ does not exist} $

I know that the second one can be disproven by paths, but I am wondering why it seems that it can be squeezed to an answer that if false. Any thoughts?