Hello everyone!

I am trying to find a limit for the 2-variavle function $\displaystyle f(x,y)=\frac{(x+y)^2}{x^2+y^2}$.

So right before changing to polar coordinates, I get something like:

$\displaystyle L=1+\lim_{(x,y) \rightarrow (+\infty ,+\infty)} \, \frac{2xy}{x^2+y^2}$$\displaystyle =1+\lim_{r \rightarrow +\infty} \, \frac{2r^2 \sin \theta \cos \theta}{r^2}$ and thus $\displaystyle L$ is bounded by 0 and 2: $\displaystyle 0<L<2$.

(1) Does this mean that the limit exists?

(2) Can we conclude that the numerator and the denomenator are of the same "order"?

(because the ratio of the numerator and the denominator is a constant)

Any help is appreciated!