I've been wondering if it's possible to define a monotonous and differentiable function that has both a horizontal and oblique assymptotes. Intuitively, seems that should be possible but I can't find a way.
- Rational functions can have either horizontal or oblique asymptotes but not both (as per this question).
- Using exponentials: The function $f(x)=x\exp(x)/\exp(|x|)$ has both horizontal asymptote ($y=0$) and oblique asymptote ($y=x$) but is not monotone.
- Another idea would be to "rotate" a hyperbola, so one of the asymptotes is sitting horizontally, and take one of the branches. I'm not sure if that's possible.