If $f$ is uniformly continuous on $[1,\infty)$ then $\exists \lim_\limits{n\to \infty}{f(x)}=L$, $-\infty \le L\le \infty$. Prove or disprove: if $f$ is uniformly continuous on $[1,\infty)$ then there exists a limit, $\lim_\limits{n\to \infty}{f(x)}=L$ where $-\infty \le L\le \infty$.
I tried to find a counterexample but it would work. I would really appreciate your help. 

Hint: $\sin \colon [1,\infty) \to \mathbb R$ is Lipschitz and hence uniformly continuous.
