The problem is this
Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n $ a $C^1$ function, and let a solution, $x(t)$ ,of $\dot x=f(x) $ be such that $$\lim_{t\to\infty} x(t)=\xi$$, then $f(\xi)=0$
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It is easy to achive this if n=1, we can use the mean value theorem, but the problem is that in higher dimension the mean value theorem can't guarantee equality. Hope you can help