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Is there any continuous function $f\colon \mathbb R\rightarrow \mathbb R$ which is bounded and also increasing and for which lim$_{x\rightarrow \infty}$ f(x) exists.

So far I only have $\sin(x)$, $\cos(x)$, $e^{-1/x}$ as examples of bounded continuous functions which dont satisfy above properties

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  • $\begingroup$ Basically all: every increasing, bounded from above function $f$ on an interval $[a,\infty)$ has a limit at $\infty$. $\endgroup$ – Taladris Nov 15 '14 at 4:48
  • $\begingroup$ Visually, you're just looking for increasing functions with two horizontal asymptopes $\endgroup$ – Alan Nov 15 '14 at 5:04
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A great example of this is $f(x) = 1 - \frac{1}{x}$. The function is increasing. And as $x\rightarrow\infty, f(x)\rightarrow 1$.

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There are many. If you want one given by a formula, you can use $\arctan x$.

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  • $\begingroup$ Please give a few more $\endgroup$ – Learnmore Nov 15 '14 at 5:01
  • $\begingroup$ What about $\frac{x^2}{1+x^2}$ for $x\ge 0$, and $-\frac{x^2}{1+x^2}$ for $x\lt 0$? For there to be a reason to produce more, there would need to be additional conditions required of the function, conditions that the two examples you now have, or their close relatives, do not meet. $\endgroup$ – André Nicolas Nov 15 '14 at 5:13
  • $\begingroup$ You are welcome. $\endgroup$ – André Nicolas Nov 24 '14 at 5:02

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