For what $\alpha$ and $\beta$ values does $\lim_{(x,y)\to (0,0)}$ $\frac {| x|^{\alpha}| y|^{\beta}}{\ x^4 + y^4 }$ goes to $0$?

Question

For what $\alpha$ and $\beta$ values does $\lim_{(x,y)\to (0,0)}$ $\frac {| x|^{\alpha}| y|^{\beta}}{\ x^4 + y^4 }$ goes to $0$ ?

Only for $\alpha >0$ and $\beta >0$

Answer 1

use that $$x^4+y^4\geq 2x^2y^2$$

Answer 2

HINT

By polar coordinates

$$\frac {| x|^{\alpha}| y|^{\beta}}{\ x^4 + y^4 } =\frac {r^{\alpha + \beta}| \cos \theta|^{\alpha}| \sin \theta|^{\beta}}{r^4( \cos^4\theta + \sin^4\theta) } =r^{\alpha + \beta-4}\frac { | \cos \theta|^{\alpha}| \sin \theta|^{\beta}}{ \cos^4\theta + \sin^4\theta}$$

then observe that $$\forall \theta \quad \cos^4\theta + \sin^4\theta\ge m>0$$