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\textbf{Partial Differentiation}

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

Explain how the function
$$f(x,y) = \frac{x^2+y^2-x^3y^3}{x^2+y^2}, \ \ \ (x,y)\ne (0,0)$$
can be defined at $(0,0)$, so that it becomes continuous at all points
of the $xy$-plane.


\textbf{Answer}

$$f(x,y) = \frac{x^2+y^2-x^3y^3}{x^2+y^2} = 1 -
\frac{x^3y^3}{x^2+y^2}$$
But
$$\left | \frac{x^3y^3}{x^2+y^2} \right | = \left |
\frac{x^2}{x^2+y^2} \right | \left | xy^3 \right | \le \left | xy^3
\right | \to 0$$
as $(x,y)\to(0,0)$.

$$\Rightarrow \lim_{(x,y) \to (0,0)} f(x,y) =1-0=1.$$

So define $f(0,0)=1$.


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