A curve that starts with exponential growth and levels out

I'm stuck on a mathematical problem in a small feature I'm building for a website. I need a function that starts out with exponential growth and than levels out as x grows. I've drawn this wonderful picture to communicate what I'm looking for since my math lingo isn't the best.

Any help appreciated.

I think you want a sigmoid function.

Perhaps you are looking for something like ${\rm erf}(x-2)+1$, where erf is the error function, ${\rm erf}(x) = \frac{2}{{\sqrt \pi }}\int_0^x {e^{ - t^2 } dt} $. You can use WolframAlpha to watch the graph: plot $f(x)={\rm erf}(x-2)+1$, say from $0$ to $5$. Further see Wikipedia's plot of the error function, here, to understand the reason for choosing ${\rm erf}(x-2)+1$.