In the figure below we use a Taylor polynomials of ln(x) and sin²(x) to determine the limit indicated in the first line.

• In the second line the natural logarithm and sine functions are replaced by Taylor polynomials each having two terms.
• In the third line x² is factored from the denominator and (1 + x + x² - 1) is replaced by x + x².
• In the fourth and fifth lines the numerator is simplified.

• In the fourth line, powers higher than x² are left off since, when they are divided by the x² in the denominator, the result will be x or a higher power of x, which will have a limit of 0 as x -> 0.

• The resulting expression is easily seem to have limit 1/2.

In the figure below we combined our previous models of population growth, with population changing any rate proportional to the population when population is low and at a rate proportional to the difference between the carrying capacity and the population when the population is high.

• Thus for low population the behavior should be of the form dP / dt = k₁ P, while for high population we expect behavior of the form dP / dt = k₂ (L - P).

• We can combine these behaviors in the same equation by letting dP / dt = k P ( L - P ).
• When P is low, L - P stays pretty near L so that k (L - P) is pretty nearly constant, so that dP / dt will be pretty nearly proportional to P.
• When P is near L, the factor L - P can undergo a significant proportional change while P doesn't change by much, so that k P remains pretty nearly constant and dP / dt is pretty nearly proportional to L - P.

• The combined equation dP / dt = k P ( L - P ) is called the logistic equation.

• The logistic equation is easily solved by separation of variables.
• We use partial fractions to perform the resulting integration, as shown below.

Using the values we found for A and B above, we rewrite the integral as in the second line below.

• In the third through the seventh lines we do the algebraic rearrange required to solve for P.
• In the fifth line C is a positive constant; in the six line, because of the + and - solutions to the absolute value equation, C becomes an arbitrary constant.
• The final function is indicated at lower right. The constant C will be evaluated for a given situation from initial conditions.

In the figure below we have divided numerator denominator by the exponential function to obtain the final form for the population function P.