$\lim_{x \to + \infty} f(x)$ when $f$ is uniformly continuous

Question

Does the $\lim_{x \to + \infty}$ f(x) of an uniformly continuous function exist? How to show it? Applying the definition of uniform continuity I cannot show that...

Answer 1

The function $\sin$ is uniformly continuous on $[0,+\infty)$, and so is the function $x\mapsto 0$. What do you conclude?

Answer 2

No. Take $f(x)=x$, which is Lipschitz, hence uniformly continuous, you get an counterexample. If you want a counterexample, which doesn't converges to $\infty$, just take $\sin(x)$ or $\cos(x)$, the Lipschitz can be proved using Mean Value Theorem.