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\textbf{Partial Differentiation}

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

Let function $f$ be given by
$$f(x,y) = \frac{\sin x \sin ^3 y}{1- \cos (x^2+y^2) }.$$
Can $f$ be defined at $(0,0)$ so that it becomes continuous there? If
it is possible, explain how.


\textbf{Answer}

$$f(x,y) = \frac{\sin x \sin ^3 y}{1- \cos (x^2+y^2) }.$$
Cannot be defined at $(0,0)$ to be continuous there.

This is because $f(x,y)$ has no limit as $(x,y)\to (0,0)$.

Observe that $f(x,0)=0$, so the limit, if it did exist would have to
be 0.

But
$$f(x,x)=\frac{\sin^4 x}{1-\cos (2x^2)} = \frac{\sin^4}{2\sin^2
(x^2)}$$
and $f(x,x)\to \frac{1}{2}$ as $x\to 0$.

So $f$ cannot be defined to be continuous at $(0,0)$.

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