Let $\alpha,\beta \in \mathbb{R}$ then find conditions for when $\lim\limits_{(x,y)\rightarrow(0,0)}\frac{x^{\alpha}y^\beta}{x^2+xy+y^2}$ exists. I don't know how to approach this exercise, I tried looking at directional paths $y=mx$ and found out that if $\alpha+\beta-2=0$ then the limit doesn't exist.
From the same directional paths I think that the condition $\alpha+\beta-2<0$ means the limit doesn't exist since if I approach $0$ from the right then the limit must be $+\infty$ and from the left is $-\infty$. Though I really don't know if I used the definition correctly, is this part right?
I'm lost at $\alpha+\beta-2>0$, I know the limit must be $0$ but I don't know how to get extra conditions so that I get a for sure convergent limit. 
 A: Often in these sorts of questions it helps to transform to polar coordinates so the limit is then of just $r \to 0$. Also, if there is anything left depending on $\theta$ (but be careful to make sure that they don't cancel out in some more complicated situations, where you may want to try a few values of $\theta$ to confirm you get different values), it usually means the limit doesn't exist as different paths will give different results.
Thus, let $x = r\cos \theta$ and $y = r\sin \theta$, and noting that $x^2 + y^2 = r^2$, the limit becomes
\begin{align}
\lim_{r \to 0} \frac{\left(r\cos \theta\right)^{\alpha} \left(r\sin \theta\right)^{\beta}}{r^2 + \left(r\cos \theta\right)\left(r\sin \theta\right)} & = \lim_{r \to 0} \frac{r^{\alpha + \beta}\left(\cos \theta\right)^{\alpha} \left(\sin \theta\right)^{\beta}}{r^2\left(1 + \left(\cos \theta\right)\left(\sin \theta\right)\right)} \\
& = \lim_{r \to 0} r^{\alpha + \beta - 2} \frac{\left(\cos \theta\right)^{\alpha} \left(\sin \theta\right)^{\beta}}{1 + \left(\cos \theta\right)\left(\sin \theta\right)} \tag{1}\label{eq1}
\end{align}
You now have $3$ basic cases to consider depending on whether $\alpha + \beta - 2$ is $\gt 0$, $= 0$ or $\lt 0$. I believe you should be able to finish the rest of it now. However, be careful with the $2$nd & $3$rd cases in terms of how the values may change depending on the path taken to the origin, as you've already determined in your question text, and with what you've written also indicates you've already got most of this figured out.
Update: As discussed in the comments below, an additional required condition fora limit to exist is that $\alpha$ and $\beta$ must both be non-negative. Also, as I later realized, as we are dealing with real powers & potentially negative values of $x$ and $y$, along with the implicit assumption that the expression in the limit is restricted to resulting in only real values, then there's the issue of for which values may there be a limit among the non-integral values of the powers. This subject is discussed in various MSE questions such as Non-integer powers of negative numbers and Dealing with non-integral powers on negative numbers, plus on StackExchange such as Calculating pow() of negative number to non-integral exponent. In general, I believe you will need to restrict the powers to be rational values where, in lowest terms, the denominator is an odd integer (e.g., $\frac{1}{3}$ and $\frac{2}{3}$ are allowed, as the cube root of a negative is defined as a negative, but $\frac{1}{2}$ is not allowed as the square root of a negative is not real), with this including integers as the denominator would be $1$. This rational number restriction is discussed further in Quora at What happens when a negative number is raised to a non-integer power?.
