Suppose that
$x(t)$ is a function defined on $[0, \infty]$ such that $x(t) \in [a, b]$ for all $t \geq 0$.
$f(x)$ is a continuous function and bounded on $[a, b]$.
$f(x(t)) \to 0$ as $t \to \infty$.
The set $f^{-1}(0)$ has only finite element $x_1, \ldots, x_n$, $x_i \in [a, b]$.
Can we prove that $x(t)$ has also finite limit as $t \to \infty$???