Over Thanksgiving dinner, Saul Griffith was complaining about the lack of mathematical literacy among people who should know better. “Take all that talk about the exponential growth of various web sites. Don’t people realize that those curves are actually sigmoidal?”
And of course, he’s right. These curves look exponential but eventually they do flatten out. In fact, one of the most important sigmoidal functions is the logistic function, originally developed to model the growth of populations.
Wikipedia notes: “The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.” In fact, most of these curves aren’t even sigmoidal, they are sinusoidal. (This is, incidentally, why Ray Kurzweil is most likely wrong about the singularity.)