Question

The  domain of the function f / g where f (x, y) = 2xy and

g(x, y) = x ²+ y² is R² .is it true or false. Give reasons for your answer

Answer 1

First of all, as $f,g$ are polynomial, $\frac{f}{g}$ are well defined on $\mathbb{R}^2 \setminus \{g=0\}$. The only point where $g(x,y)=0$ is $x=y=0$. Thus, the function is not defined on $(0;0)$. In addition, we can not extend $f/g$ to (0,0) in any natural way, as the limit $\lim_{x,y\to 0} \frac{f(x,y)}{g(x.y)} = \lim_{r\to 0} \frac{2 r^2 \sin \theta \cos \theta}{r^2}$ does not exist (as it depends on $\theta$). Therefore $\frac{f}{g}$ is not defined on $\mathbb{R}^2$ and can't be extended in any natural way.