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It's in the title. I am looking for simple function that passes through 0, square root like, never reaches 100, but comes closer and closer to it.

I'm sure this is very basic, nothing fancy. But I don't know it now.

Thanks a lot!

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What do you mean when you say "square root-like"? Do you mean that you need it to have infinite slope at $x=0$? Do you want $f(x)\approx \sqrt{x}$ when $x$ is small? Or do you just want a function whose slope decreases and which tends to 100 as $x$ tends to infinity? –  Chris Taylor Oct 4 '12 at 17:07
    
My question aims at: "whose slope decreases and which tends to 100 as x tends to infinity." I am basically trying to give a mathematical impression of a organism that recovers after a shock (0): First very fast, then slower and slower. –  Frank Oct 4 '12 at 17:14
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2 Answers 2

up vote 3 down vote accepted

What about $ x \mapsto \frac{100}{\pi/2}\arctan $?

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This function do not expose the vertical tangent at $x=0$ feature of $\sqrt x$. –  enzotib Oct 4 '12 at 17:01
    
I marked this answer as solution because of its simplicity. –  Frank Oct 4 '12 at 17:24
    
I felt that the vertical tangent is not so important, and he was looking for these 6 characters: $\arctan$ –  Berci Oct 5 '12 at 10:39
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There are infinite solutions to your request, one can be

$$ f(x)=100\frac{\sqrt x}{1+\sqrt x} $$

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