Is a Limit Point Always an Equilibrium in Differential Equations?

[simo1]

Say x is a solution of the DE x’=f(x) and f is a continuous derivative on its domain,
if lim┬(t→͚inifinity⁡〖x(t)〗=p then p is equilibriumhow can I show that this is true

 

[chisigma]

Say x is a solution of the DE x’=f(x) and f is a continuous derivative on its domain,
if lim┬(t→͚inifinity⁡〖x(t)〗=p then p is equilibriumhow can I show that this is true

I suppose You mean that p is 'equilibrium' if f(p)=0. In this case if $\displaystyle \lim_{t \rightarrow \infty} x(t)= p$ then given $\varepsilon >0$ it exists a $t_{0}$ so that for any $t>t_{0}$ is $|x(t) - p|< \varepsilon$. Consequence of that is that for any $t_{1}> t_{0}$ and $t_{2}>t_{0}$ is $|x(t_{1}) - x(t_{2})| < 2\ \varepsilon$, so that is $\displaystyle \lim_{t \rightarrow \infty} x^{\ '} (t) = 0$... Kind regards $\chi$ $\sigma$