Help calculating the limit of an Integral defined in R^2

Question

Calculate $$\lim_{k\to\infty}\int_{\mathbb R^2}\frac{(x^2+y^2)^{k/2}}{1+(x^2+y^2)^{(k+3)/2}}\,dx\,dy$$ without using the change of variable theorem, my idea was to used the Dominated Convergence Theorem but I get stuck calculating the limit, thanks in advance!

Answer 1

Hint:

$$\displaystyle\lim_{k \to \infty} \int_{x^2+y^2 < 1}\frac{(x^2 + y^2)^{k/2}}{1 + (x^2 + y^2)^{(k+3)/2}} \, dx \, dy = 0$$

$$\displaystyle\lim_{k \to \infty} \int_{x^2+y^2 > 1}\frac{(x^2 + y^2)^{k/2}}{1 + (x^2 + y^2)^{(k+3)/2}} \, dx \, dy = \int_{x^2+y^2 > 1}\frac{1}{(x^2 + y^2)^{3/2}} \, dx \, dy$$

The dominating functions should be obvious.