The limit of $u(x,y)$ when $x^2+y^2\to\infty$ I cannot understand what the following does it mean:$\lim\limits{u(x,y)}=0$ , when $x^2+y^2\to\infty$? Does it mean that at least one of the variables goes to $\infty$ or both, or something else?
 A: I would not focus on the meaning of $x^2 + y^2 \to \infty$ in isolation.
The statement $\lim_{x^2 + y^2 \to \infty}u(x,y) = L$  means for any $\epsilon > 0$ there exists $K > 0$ such that $|u(x,y) - L| < \epsilon $ for all $(x,y)$ where $x^2 + y^2 > K$.
A: That sounds like it means at least one of the variables goes to $\infty$. $x^2 + y^2 = r^2$ is often times used to describe a circle (centered at the origin) with radius r. If $x^2 + y^2 \to \infty$, then you are considering a circle (centered at the origin) with infinite radius. Do you have any information on $u(x,y)$?
A: Consider that $x^2+y^2=\|(x,y)\|_2$.
So the meaning of
$$
x^2+y^2\to\infty
$$
is, that you consider something for points with large euclidean norm. If the euclidean norm goes to infinity, your points are getting more and more far distant to the orgin.
Even if the point is far from the orgin, the coordinates don't have to be both large. You can even conclude $x^2\to\infty$ or $y^2\to\infty$.
Consider
$$
x_n=(1-(-1)^n)n\text{ and }y_n=(1-(-1)^{n-1})n.
$$
You see $x_n\not\to\infty$ and $y_n\not\to\infty$ but
$$
x_n^2+y_n^2=4n^2\to\infty.
$$
