TITLE: What kind of growing function has a constant as limit?
QUESTION [5 upvotes]: My knowledge in mathematics are a bit old and I'm looking for functions with constant as limit. The function must always grow.
The curve should be something similar to $\sqrt{x}$ or $\ln(x)$ but with $\lim _{x\to \infty \:}f\left(x\right)=\mathrm{constant}$
Could you please help to find this kind of functions?

REPLY [3 votes]: If you just want to have some functions with the asked properties, you can easily construct one as Xoff pointed out. If you are looking for a class of functions (possibly from applied mathematics), the logistic function $$f:\mathbb R\rightarrow\mathbb R,~f(x)=\frac{L}{1+e^{-k(x-x_0)}}$$ might be something you are looking for (for the meaning of the constants $L,k,x_0$ consider the wikipedia page, it has lots of information on this). 
If your familiar with stochastics and random variables, you could take a look at the cumulative distribution function. It is monotonicically non-decreasing with $\lim\limits_{x\to\infty}f(x)=1$.