For $\alpha>0$ and $R=[1,+ \infty) \times [1,\infty) $
Compute, if the limit exist:
$$\lim_{x^2+y^2\rightarrow\infty} \frac{xy^{\alpha}}{1+x^2+x^2y^2} $$ with $(x,y)\in R$
I managed to prove that
1)If the limit exist, the limit is $0$
2)The limit is $0$ for $0 < \alpha < 1$ (using polar cordinates)
3)The limit doesn't exist for $\alpha >2$ (evaluating the limit $\lim_{t->\infty }f(t,t^k) $ where k is a generic exponent >0)
The problem is that i can't prove anything for $1 < \alpha <2$
How can i resolve it?