My point was that fertility rates or birth rates have nothing to do with population growth, except for a longer or shorter time frames (doubling time). These rates ignore death rates. You need both in order to establish population growth or decline. If death rates exceed birth rates the population will decline . If birth rates exceed death rates the population will grow, until it reaches unsustainable numbers. Then birth rates and fertility rates will reach a forced zero or we have a nice big war or a pandemic. I appreciate your attention to detail. But in your zeal to correct me you miss the IMPORTANT point about the exponential function, which was the subject of my post. In the long run, birth rates and death rates will reach zero, and it will do so by our choice or by nature's choice. That is the dilemma at hand!