Suppose that $\liminf_{t\rightarrow\infty}\|x(t)\|\neq0$, so there exists a neighborhood $H$ of 0 and an increasing sequence $(t_n)$ such that $x(t_n)\not\in H$. Let $V$ be a neighborhood that the stability suggests for $H$. By the definition, $x(t_n)\not\in V\cap f^{-1}(V)$. Hence, $f(x(t_n))\not\in V$ for all $n$, whence 
$\liminf_{t\rightarrow\infty}f(x(t))\neq0$.