# Is it possible to define a monotonous and differentiable function with horizontal and oblique asymptotes?

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.
• Your third idea is the one which works. – Crostul Oct 27 '16 at 8:43
• $$f(x)=\frac 12 (\sqrt{x^2+1}+x)$$ – Crostul Oct 27 '16 at 8:45
• @Crostul, could you post this as an answer? – FKaria Oct 27 '16 at 9:03

$$f(x)= \frac 12 (\sqrt{x^2+1}+x)$$
$$f(x) = \begin{cases}e^x & x\leq 0\\ x + 1 & x\geq 0\end{cases}$$