Simple counterexample for if $\lim_{x\to\infty}f(x)=\infty$ then $f$ is monotonic

If $$\lim_{x\to\infty}f(x)=\infty,$$ then we can't conclude that there exists an $M$ where $f$ is monotonic on $[M,\infty)$ because $f(x)=\ln x+\sin x$ disproves that claim. What I want to know is, if my example is the "standard" counterexample, or maybe there is a more "simple" counterexample.