limit of multivariable function as x,y approach to infinity

Question

can i solve this limit using polar coordinate? $$\lim_{(x,y)\to\infty} \frac{x^2+y^2}{x^2+(y-1)^2}=$$ $$\frac{r^2}{r^2-2r\sin\theta +1}=\frac{1}{1-\frac{2\sin\theta}{r}+\frac{1}{r^2}}=1$$

Answer 1

Yes of course your solution is fine, as an alternative by $x=u$ and $y-1=v$

$$ \frac{x^2+y^2}{x^2+(y-1)^2}=\frac{u^2+(v+1)^2}{u^2+v^2} =1+\frac{1}{u^2+v^2}+\frac{2v}{u^2+v^2} \to 1$$

for the latter, in order to avoid cases, we can use that by AM-QM

$$\frac{2|v|}{u^2+v^2} \le \frac{2(|u|+|v|)}{u^2+v^2} \le \frac{2\sqrt 2}{\sqrt{u^2+v^2}} \to 0$$