limit of multivariable function as x,y approach to infinity

Mathematics Asked by simon on November 26, 2020

can i solve this limit using polar coordinate?
$$lim_{(x,y)toinfty} frac{x^2+y^2}{x^2+(y-1)^2}=$$
$$frac{r^2}{r^2-2rsintheta +1}=frac{1}{1-frac{2sintheta}{r}+frac{1}{r^2}}=1$$

One Answer

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

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

Correct answer by user on November 26, 2020

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