Show that $ \lim_{(x,y) \to 0} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \alpha/\gamma + \beta/\delta > 1.$ Ted Shifrin on this site posed an interesting problem to me: show that 
$$ \lim_{(x,y) \to (0,0)} \frac {|x|^ \alpha |y|^ \beta} {|x|^ \gamma + |y|^ \delta} \text {exists} \iff \frac\alpha\gamma + \frac\beta\delta > 1, \,\,\,\,\text{where } \alpha, \beta, \gamma, \delta >0$$
I think I've got the $(\Leftarrow)$ direction as follows: assume WLOG that $\gamma \leq \delta$. Then switch to polar coordinates and get 
$$\frac {r^ {\alpha + \beta} (*) } {r^\gamma (1(*) + r^{\delta - \gamma}(*))},$$
where $(*)$ represents some trig stuff that is bounded near zero. 
Edit: I need to make sure that $(1(*) + r^{\delta - \gamma}(*))$ is bounded below here.
The other direction is giving me some trouble. It seems we need a lower bound for the fraction (something to force to zero), and I'm not sure where to find one. In particular, I'm not sure what to do with the denominator. Any ideas?
 A: For the $(\Leftarrow)$ direction, we first note that
$$\lim_{(x,y) \to 0} \frac{|x|^\alpha |y|^\beta}{|x|^\gamma + |y|^\delta} = \lim_{r_1,r_2 \to 0^+} \frac{r^\alpha_1 r^\beta_2}{r^\gamma_1 + r^\delta_2} = \lim_{t_1,t_2 \to 0^+} \frac{t^{\alpha\delta}_1 t^{\beta\gamma}_2}{t^{\gamma\delta}_1 + t^{\gamma\delta}_2}, $$
since a sequence $(t_n, s_n) \to (0,0)$ if and only if $(t^\delta_n, s^\gamma_n) \to (0,0)$
Now for the rightmost limit, your basic argument works: by switching to polar coordinates we have
$$\lim_{r \to 0} \frac{r^{\alpha \delta + \beta \gamma}\left|\sin(\theta)^{(\beta \gamma)} \cos(\theta)^{(\alpha \delta)}\right|}{r^{\gamma \delta} \left(\left|\sin(\theta)^{(\gamma \delta)}\right| + \left|\cos(\theta)^{(\gamma \delta)}\right|\right)}, $$
which we can easily show to converge (to zero) if $\alpha \delta + \beta \gamma > \gamma \delta$.
For the ($\Rightarrow$) direction, assume that the said limit exists. Then, in particular, this limit also exists:
$$\lim_{t \to 0^+} \frac{(t^\delta)^\alpha (t^\gamma)^\delta}{(t^\delta)^\gamma + (t^\gamma)^\delta},$$
which clearly exists only if $\alpha \delta + \beta \gamma > \gamma \delta$.
EDIT: The last limit exists if $\alpha \delta + \beta \gamma \geq \gamma \delta$.
To show that we can't have $\alpha \delta + \beta \gamma = \gamma \delta$, notice that while in this case the above limit is $\frac12$, we also have other "subsequences" converging to zero: let $N$ be such that $\beta N > \gamma$ and $\delta N > \gamma$. Then
$$ \frac{t^\alpha (t^N)^\beta}{t^\gamma + (t^N)^\delta} \approx \frac{t^\alpha (t^N)^\beta}{t^\gamma} \to 0 $$
Thus the limit does not exist when $\alpha \delta + \beta \gamma = \gamma \delta.$
