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\begin{center}
\textbf{Partial Differentiation}

\textit{\textbf{Limits}}
\end{center}

\textbf{Question}

Evaluate the given limit. If the limit does not
exist, explain why.

$$ \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{2x^4+y^4}$$


\textbf{Answer}

\begin{eqnarray*}
\textrm{If } x & = & 0 \textrm{ and } y\ne 0\\
\Rightarrow \frac{x^2y^2}{2x^4+y^4} & = & 0\\
\textrm{If } x & = & y \ne 0\\
\Rightarrow \frac{x^2y^2}{2x^4+y^4} & = &
\frac{x^4}{2x^4+x^4}=\frac{1}{3}
\end{eqnarray*} 

$\Rightarrow \ds \lim_{(x,y)\to(0,0)} \frac{x^2y^2}{2x^4+y^4}$
Does not exist.


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