Chapter 3.1, Problem 60E

Solution

To decide: if the given statement is true or false.

True

Given information:

The statement, “Every logistic growth function has two horizontal asymptotes.”

Definition Used:

Logistic Growth Function:

Let a , b , c , and k be positive constants, with $b<1$ . A logistic growth function in x is a function that can be written in the form

  $f\left(x\right)=\frac{c}{1+a\cdot b^x}\text{ or }f\left(x\right)=\frac{c}{1+a\cdot e^{-kx}}$

Where the constant c is the limit to growth.

Explanation:

By the above definition, the numerator of a logistic growth function is known as its limit to growth. Since, for any logistic growth function  f ( x ) ,

  $\underset{x\to \infty }{\lim }f\left(x\right)=c$

and thus  $y=c$  is a horizontal asymptote.

Also, every logistic growth function will approach zero as  x  tends to negative infinity, that is, for a logistic growth function  f ( x ),

  $\underset{x\to -\infty }{\lim }f\left(x\right)=0$

and thus  $y=0$ is a horizontal asymptote.

Thus, the given statement is true.