How to prove that $\lim\limits_{(x,y) \to (0,0)} \frac{\left | x \right |^{\frac{3}{2}}y^{2}}{x^{4} + y^{2}} \rightarrow 0$ How can I prove that
$$\lim_{(x,y)\to (0,0)} \frac{\left | x \right |^{\frac{3}{2}}y^{2}}{x^{4} + y^{2}} \rightarrow 0\;?$$
Thanks!

For $y \neq 0$,
$$
0 \le \frac{{|x|^{3/2} y^2 }}{{x^4  + y^2 }} = \frac{{|x|^{3/2} }}{{x^4 /y^2  + 1}} \le \frac{{|x|^{3/2} }}{1} = |x|^{3/2}  \to 0.
$$
EDIT: In retrospect, simply note that $0 \le \frac{{y^2 }}{{x^4  + y^2 }} \le 1$, for $(x,y) \neq (0,0)$, to conclude that $\frac{{|x|^{3/2} y^2 }}{{x^4  + y^2 }} \to 0$ as $x \to 0$, independently of $y$.
