Show that the equilibrium point $\displaystyle x_0=0$ for the differential equation $\displaystyle x^{\prime}=0$ is stable but not asymptotically stable.
In this problem you can actually integrate and find the solution
$\displaystyle x(t) = x_0$
Now, with $\displaystyle x_0 = 0$ gives your solution. Now perturb it so
$\displaystyle x^{*}(t) = x_1$.
So,
$\displaystyle | x(t) - x^{*}(t)| = x_1 $ for all t (this a stable)
However, we don't have for some $\displaystyle \delta > 0$
$\displaystyle | x(t) - x^{*}(t)| \le \delta \; \text{for}\; t > t*$ (this a asymptotic stability).