Looking for a function with $f(0) = 0$, $f'(0) = 1$ and $\lim\limits_{x\to\infty}f(x)=1$ I'm looking for a monotonic, continuous function function with these properties:
$$f(0) = 0$$
$$f'(0) = 1$$
$$\lim\limits_{x\to\infty}f(x)=1$$
I also need a coefficient that allows me to configure the speed at which it reaches 1.
I've got the feeling that there is a well-known function that I am looking for but my brain just refuses to give it to me right now.
 A: $f(x) = 1-e^{-x}$ would seem to be a candidate. 
Another one is
$$f(x) = 1-e^{-(e^x-1)}$$
In general:
$$f(x) =  1-e^{-g(x)}$$
where $g(0)=0$, $g'(0)=1$, and $\lim_{x \to \infty} g(x) = \infty$.
A: I suggest $$f(x)=\frac{x\alpha}{x\alpha+1},\qquad\alpha>0.$$
A: Here are a variety of functions (taken from http://en.wikipedia.org/wiki/Sigmoid_function)!

Many of these types of equations can be parameterized to control rate of convergence. A simple way would be to paramaterize a combination of two of these functions e.g. $f(x) = ag_1(x) + (1-a)g_2(x)$, where $g_1,g_2$ are relatively "fast" or "slow" converging functions.
A: This might make things easier. Take any function such that $f(0)=0, f'(0)=1$. Then
$$\frac{f(x)}{1+f(x)}$$
automatically satisfies both of these and your third requirement if $\lim_{x\to\infty} f(x)=\infty$. It's also monotonically increasing when your original function is. If you have any single function $g$ such that $g(x)=0, g'(x)=1$, then $\frac 1n g(nx)$ will also have these properties, allowing you to generate a family of functions. For example, if we take $f(x)=\sinh(x)$, then our family of functions is $\frac 1n\sinh(nx)$ and our overall set of solutions is $$\frac{\sinh(nx)}{n+\sinh(nx)}$$
which you can verify have all of your needed properties and approach $1$ at varying rates. Plug in any increasing function with $f(x)=0$ and $f'(x)=1$ in place of $\sinh(x)$ and you'll have all the fine-tuning you need.
A: $$f(x) = \left\{\begin{array}{l l} x & x\le 1\\ 1 & x > 1\end{array}\right.$$
