\displaystyle $\begin{align*}
 \left| \frac{x^3 y^2 - 2 y^4}{\sqrt{2 x^2 + y^4}} \right| = \frac{| x^3 y^2 - 2 y^4 |}{\sqrt{2 x^2 + y^4}} \leq \frac{| x^3 y^2 | + | 2 y^4 |}{\sqrt{y^4}} = \frac{ |x|^3 y^2 + 2 y^4 }{y^2} = |x|^3 + 2 y^2 < \delta^3 + 2 \delta^2 = \varepsilon
\end{align*}$