Given the function $$f_a(x,y)=\begin{cases}y^\alpha \frac{\sin(xy)}{x^2+y^2}\ \ \ \ \text{ for }(x,y)\neq (0,0)\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ for }(x,y)=(0,0)\end{cases}$$
with $\alpha=1,2,3,\dots$, for which values of $\alpha$ is the function continuous on the plane?
So I tried going with the fact that if it has to be continuous then $$\lim_{(x,y)\to(0,0)}y^\alpha \frac{\sin(xy)}{x^2+y^2}=0$$ so I tried approaching with polar coordinates: $$\lim_{r\to 0} r^\alpha \sin^\alpha \theta\frac{\sin(r^2 \cos\theta\sin\theta)}{r^2}=\lim_{r\to 0}\sin^\alpha \theta\frac{\sin(r^2\cos\theta\sin\theta)}{r^{2-\alpha}}$$ so then I concluded that if $\alpha\geq 2$, the limit would be $0\ \forall \alpha\in\mathbb{R}$ hence would be always continuous and then continued with $\alpha<2$ with the limit and doing l'hôpital maybe but it doesn't seem to be ending anywhere.