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I'd like to say logarithmically decreasing but it does not have to decrease to zero.

An example of an increasing function which flattens out at around 4.5%.

enter image description here

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It still isn't clear what "flatten out" means precisely. Do $y=\sqrt{x}$ or $y=\log(x)$ "flatten out"? These functions have the property that they are always increasing, do not tend to a finite limit, but their rate of increase goes to $0$ as $x$ goes to infinity. – Jonas Meyer Feb 8 '11 at 0:10
@Jonas: No, they don't "flatten out". Horizontal asymptote is exactly what I was looking for. – Jacob Feb 8 '11 at 2:29
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Thanks! Just wanted to be sure. – Jonas Meyer Feb 8 '11 at 3:24

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I don't know what "flattens out" means (and your image doesn't load), but maybe you're thinking of functions of a variable $x$ which tend monotonically to a limit as $x$ tends to infinity. Having a "horizontal asymptote" is another common related term.

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The function is also concave, perhaps even log-concave. – Yuval Filmus Feb 7 '11 at 23:04
The image problem might be fixed. And that sounds about right. – Jacob Feb 7 '11 at 23:14

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