Some populations initally grow exponentially but eventually level off. Equations of the form

where M, A, and k are positive constants, are called logistic equations and are often used to model such populations. (We will investigate these in detail in Chapter 9.) Here M is called the carrying capacity and represents the maximum population size that can be supported, and A = M - P₀/P₀, where P₀ is the initial population.

(a) Compute lim _{t →∞} P(t). Explain why your answer is to be expected.
(b) Compute lim _{M →∞} P(t). (Note that A is defined in terms of M.) What kind of function is your result?

P(t) 1 + Ae -kt