<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
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\begin{document}
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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 &amp; = &amp; 0 \textrm{ and } y\ne 0\\
\Rightarrow \frac{x^2y^2}{2x^4+y^4} &amp; = &amp; 0\\
\textrm{If } x &amp; = &amp; y \ne 0\\
\Rightarrow \frac{x^2y^2}{2x^4+y^4} &amp; = &amp;
\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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