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\textbf{Partial Differentiation}

\textit{\textbf{Limits}}
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\textbf{Question}

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

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


\textbf{Answer}

If $\ds f(x,y)=\frac{x}{x^2+y^2}$

Then $|f(x,0)| = |1/x| \to \infty$ as $x\to 0$.

But $|f(0,y)|=0\to 0$ as $y\to 0$.

And so $\ds \lim_{(x,y)\to(0,0)}f(x,y)$ does not exist.


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