Let $a,b,c,d$ be positive real numbers. Show that the limit $$\lim_{(x,y) \to (0,0)}\frac{|x|^{a}|y|^{b}}{|x|^{c} + |y|^{d}} = 0$$ if only if $$\frac{a}{c} + \frac{b}{d} > 1.$$
This question seems to be basically about algebraic manipulation, but I cannot seem to find a good way. Can anybody help me?
Using substitutions $x\to x^{1/c}$ and $y\to x^{1/d}$ the problem reduce to $$\lim_{(x,y) \to (0,0)}\frac{|x|^r|y|^s}{|x| + |y|} = 0$$ iff $r+s>1$ where $r=\dfrac{a}{c}$ and $s=\dfrac{b}{d}$. With polar substitutions $x=\rho\cos\theta$ and $y=\rho\sin\theta$ we have $$\lim_{(x,y) \to (0,0)}\frac{|x|^r|y|^s}{|x| + |y|} = \lim_{(\rho,\theta) \to (0,0)}\frac{\rho^{r+s-1}(|\cos\theta|^r|\sin\theta|^s)}{|\cos\theta|+|\sin\theta|} = 0$$ iff $r+s>1$.