Limit of the two variable function $f(x,y).$ How to show that $$\lim_{(x,y)\to(0,0)}\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}=0$$ I tried with different paths as $x=0,y=0, y=x$ its comes to zero but i have no general idea. Please help. Thanks to lot.

You have 
\begin{align}
\left| \frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}\right|
&=\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}
\leq
\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}\,(y^{2})}\\ \ \\
&=\frac{x^2}{\sqrt{x^2+y^2}}\leq\frac{x^2}{\sqrt{x^2}}=|x|\\ \ \\
&\leq\sqrt{x^2+y^2}
\end{align}
