<s>[INST] Is the limit $\lim_{(x,y) \to \infty} \frac{x+2y}{x^2 - 2xy + 2y^2}$ zero? I have this limit:$$\lim_{(x,y) \to \infty} \frac{x+2y}{x^2 - 2xy + 2y^2}$$
At first sight seems that limit equals 0. But WolframAlpha says that there is no limit. I tried to prove it. I considered cases $y = kx$, and so on. I never got to find subsequence, whose has limit $\neq 0$. 
I think there is a problem in denominator. When $x\rightarrow \infty$ and $y \rightarrow \infty$ we got there $\infty - \infty + \infty$. It's unclear what to do with it and how to find necessary subsequence.
Maybe i'm on the wrong way to solve it. Please, give me a tip.
 [/INST] $$
\begin{align}
(x-y)^2+y^2
&=x^2-2xy+2y^2\\
&=(2-\phi)x^2+(\phi-1)x^2-2xy+\phi y^2+(2-\phi)y^2\\
&=(2-\phi)\left(x^2+y^2\right)+\left(\sqrt{\phi-1}\,x+\sqrt{\phi}\,y\right)^2\\
&\ge(2-\phi)\left(x^2+y^2\right)\\
&=\frac1{\phi^2}\left(x^2+y^2\right)
\end{align}
$$
Therefore,
$$
\begin{align}
\left|\frac{x+2y}{(x-y)^2+y^2}\right|
&\le\phi^2\frac{|x+2y|}{x^2+y^2}\\
&\le\frac{3\phi^2}{\sqrt{x^2+y^2}}
\end{align}
$$
</s>