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

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  • $\begingroup$ I accept this answer as it is defined in a single equation and second order differentiable $\endgroup$ – FKaria Oct 27 '16 at 9:08
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$$f(x) = \begin{cases}e^x & x\leq 0\\ x + 1 & x\geq 0\end{cases}$$

does the trick just fine, I think.

You could also do the "rotate a hyperbola" trick as well, to get a "nicer" function that is not piecewise defined. But the quickest answer from the top of my head is the one above.

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