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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.

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


\textbf{Answer}

Let $\ds f(x,y)=\frac{\sin(xy)}{x^2+y^2}$.

\begin{eqnarray*}
\Rightarrow f(0,y) & = & 0/x^2 = 0 \to 0\\
\textrm{as } x\to 0 & &\\
\textrm{But } f(x,x) & = & \frac{\sin x^2}{2x^2} \to \frac{1}{2}\\
\textrm{as } x\to 0 & & 
\end{eqnarray*}

$\Rightarrow \ds \lim_{(x,y)\to(0,0)} \frac{\sin(xy)}{x^2+y^2}$

Does not exist


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