Two-variable limit question $\lim\limits_{(x,y)\rightarrow (0,0)} \dfrac{x^2y^2}{(x^2+y^4)\sqrt{x^2+y^2}}$
How to solve this two-variable limit? Thanks :D

Let
\begin{equation}
f(x,y)=\frac{x^2y^2}{(x^2+y^4)\sqrt{x^2+y^2}}
\end{equation}
If $y=0$, $\lim_{(x,y)\rightarrow(0,0)}f(x,y)=0$. If $y\neq0$, say approach the $(0,0)$ do not along with x-axis, we have:
\begin{equation}
f(x,y)=\frac{x^2y^2}{(x^2+y^4)\sqrt{x^2+y^2}}\leq\frac{x^2y^2}{2|x|y^2\sqrt{x^2+y^2}}=\frac{|x|}{2\sqrt{x^2+y^2}}\\
0\leq\lim_{(x,y)\rightarrow(0,0)}f(x,y)\leq\lim_{(x,y)\rightarrow(0,0)}\frac{|x|}{2\sqrt{x^2+y^2}}=0
\end{equation}
In summary, $\lim_{(x,y)\rightarrow(0,0)}f(x,y)=0$.
