A function $f$ such that $f(x)$ increases from $0$ to $1$ when $x$ increases from $0$ to infinity? 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.  


Simple version:
  $$f(x)=1-\mathrm e^{-a\sqrt{x}}\qquad (a\gt0)$$

Slightly more elaborate version:$$f(x)=1-\mathrm e^{-a\sqrt{x}-bx}\qquad (a\gt0,\ b\geqslant0)$$
Every such function fits every requisite in the question, including the infinite slope at $0$. The parameter $a$ can help to tune the increase near $0$. The parameter $b$ can help to tune the increase near $+\infty$. To get even quicker convergence to $1$ when $x\to+\infty$, one can replace $-bx$ in the exponent by $-bx^n$ for some $b\gt0$ and $n\gt1$.
Once one understands the principle, a host of other solutions springs to mind.
