I, too, thought it was related. It is not.

Limits to Growth, whether you agree with their conclusions or not, is an actual model.

The google docs above is one guy fiting a curve to a bunch of demographic statistics. There is no real causality there, no attempt to make an interpretation of the numbers. If he would have done so, he would - as others have already pointed out - reached the conclusion that: "Since birth rates decrease when people's lives improve, Progress will make our life so fulfilling that we will stop having children altogether. Then we will live happily ever after until we all die of old age."

Well, one thing more you could say is whether you're pointing this out as an example to emulate or avoid. :)