Example of a semimartingale with special properties

Question

I have to find an example of a semimartingale X such that $\lim_{t \rightarrow \infty} X_t$ exists a.s. and $X$ is not a semimartingale up to infinity.

I think it could be a deterministic function of finite variation. But I'm not sure what the exact function looks like.

Any ideas?

Thanks