Assume $f$ is uniformly continuous and unbounded on $[0,\infty)$. Whether the following statement is true? Assume $f$ is uniformly continuous and unbounded on $[0,\infty)$. Is it true that either $\displaystyle\lim_{x\to\infty}f(x)=+\infty$ or $\displaystyle\lim_{x\to\infty}f(x)=-\infty$ 
Please give a hint for the problem. I have tried to find some oscillating function, but could not find. Give some hint for this problem.

Intuitively, $x\sin(x)$ was on the right the track but the function has to keep forcing itself to $0$ every $\pi$, so if only there was a way to slow down the oscillation as the function grows unbounded. 
For a pretty counterexample, use 
$$f(x) = \begin{cases} x\sin(\log|x|) & x\neq 0 \\ 0 & x=0 \\ \end{cases}$$
The derivative $\cos(\log|x|) + \sin(\log|x|)$ is bounded, making this function Lipschitz, hence uniformly continuous, but its limit as $x\to \infty$ does not exist.
