Whether does the number e come from!?!?

Suppose the number of bacteria, n 0, in a dish doubles in unit time. If a very simple growth model is adopted, you can think of the number of bacteria remaining constant during the period and then, at t = 1, the number doubles instantaneously. … but this is not very realistic

To make it more realistic, suppose that the number of bacteria remains constant during the periods 0≤t<0.1, 0.1 ≤t<0.2, 0.2 ≤t<0.3, …, 0.9 ≤t<1 and the number of bacteria grows by 10% (one tenth) at times t=0.1, t=0.2, t=0.3, …, t=0.9, t=1

Then, when t=1, the number of bacteria is *Note that the number has more than doubled. In fact, it has increased by a factor of almost 2.6 because of the compounding effect.

Ultimately, the smaller the length of the time interval, the more accurate the growth model is. For example, if the number of bacteria remains constant during the periods 0≤t<0.01, 0.01 ≤t<0.02, 0.02 ≤t<0.03, …, 0.99 ≤t<1 and the number of bacteria grows by 1% (one hundredth) at times t=0.01, t=0.02, t=0.03, …, t=0.99, t=1

Then what is the formula for the number of bacteria?