TITLE: A function $f$ such that $f(x)$ increases from $0$ to $1$ when $x$ increases from $0$ to infinity?
QUESTION [32 upvotes]: I am looking for a function f(x) with a value range of [0,1].
f(x) should increase from 0 to 1 while its parameter x increases from 0 to +infinity.
f(x) increases very fast when x is small, and then very slow and eventually approach 1 when x is infinity.
Here is a figure. The green curve is what I am looking for:

Thanks.
It would be great if I can adjust the slope of the increase. Although this is not a compulsory requirement.

REPLY [16 votes]: I'm surprised no one has mentioned erf(x) aka the "Error Function", defined roughly as the normalized area under the bell curve as a function of the upper limit of integration. For x > 1, it satisfies your requirement and also has a slope that is easily controlled (and made arbitrarily large at 0) by the width of your Gaussian.