<s>[INST] Evaluating the limits $\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$ and $\lim_{(x,y)\to(\infty,8)}(1+\frac{1}{3x})^\frac{x^2}{x+y}$ I got the following problem:
Evaluate the following limits or show that it does not exist: 
$$\lim_{(x,y)\to(\infty,\infty)}\frac{2x-y}{x^2-xy+y^2}$$
and
$$\lim_{(x,y)\to(\infty,8)}\left(1+\frac{1}{3x}\right)^\frac{x^2}{x+y}$$
I tried for an hour and half evaluating each of those limits but I failed and I got nothing useful to share.
Some hints will be appreciated.
Thanks.
 [/INST] By setting:

$$x:=r\cos\theta \\ y:=r\sin\theta$$

You can turn: $$\lim_{\substack{
x\rightarrow\infty\\
y\rightarrow\infty}}
{f(x,y)}$$ into: $$\lim_{r\rightarrow\infty}{g(r,\theta)}.$$
So that:

$$ \frac{2x-y}{x^2-xy+y^2}=\frac{1}{r}\cdot\frac{2\cos\theta-\sin\theta}{(1-\cos\theta\sin\theta)}\,\overset{r\rightarrow\infty}{\longrightarrow}\,0$$

The second one is easier. By noting that:
$$
\lim_{\substack{
x\rightarrow\infty\\
y\rightarrow 8}}{\frac{y}{x}} = 0
$$
We get:

$$ \left(1+\frac{1}{3x}\right)^{\frac{x^2}{x+y}}=\left(\left(1+\frac{1}{3x}\right)^{3x}\right)^{\frac{1}{3+3\frac{y}{x}}} \longrightarrow e^{\frac{1}{3}}$$

</s>