Which values of $\alpha, \beta > 0$ make $f(x,y)=\frac{|{x}|^{\alpha}|{y}|^{\beta}}{x^2 + y^2}$ continuous?
Multivariable limit proof: $\lim\limits_{(x,y)\rightarrow (0,0)}\frac{\left|x\right|^a\left|y\right|^b}{\left|x\right|^c + \left|y\right|^d} = 0$