Suppose we have an initial value problem $$\dot{x}=f(x),$$ $$x(0)=x_0$$ where $f\in C^1(E)$ for some open $E$. Moreover, suppose we have a solution $x(t)$ such that $$\underset{t\rightarrow\infty}{\lim}x(t)=X$$ for $x\in E$. The goal is to show that $f(X)=0$.
My initial idea is taking a limit of the differential equation, because continuity of $f$ gives $f(X)$ in the right-hand side. The problem I have with this approach is applying the limit to the left-hand side. I have also heard Mean Value Theorem can help with this but I don't see how. Anyone have guidance for this? Thanks in advance.
I believe you are correct about the mean value theorem, but I will, effectively, be using an integral form instead of the differential form.
The average rate of change is easier to work with than the derivative for precisely the limit reason you mentioned. Integrating the equation and dividing, we recover the average value $$ \frac{x(b)-x(a)}{b-a}=\frac{1}{b-a}\int_a^bf(x(t))dt. $$ Extract $f(X)$: $$ \frac{x(b)-x(a)}{b-a}=\frac{1}{b-a}\int_a^b[f(x(t))-f(X)]dt+f(X). $$ Now put $b=2a$ and send $a\to\infty$. The left hand side vanishes, and the first term on the right hand side vanishes if we apply the integral mean value theorem to the continuous function $f$: $$ \frac{1}{a}\int_a^{2a}[f(x(t))-f(X)]dt=f(x(s))-f(X), $$ for some $s\in[a,2a]$. This difference is small for all such $s$ if $a$ is large.