Many times we are concerned with the “end behavior” of a function. That is, what does a function do as x approaches infinity. This becomes very important when something is a function of time, and we want to know the behavior as time extends into the future. Let’s look at our turkey population model from a previous assignment.

The population of a flock of turkeys is modeled by the function: Let’s look at the graph of the function

P, number of turkeys t, in years The population peaks around the 3 year mark and then seems to level off. By examining the limit as t approaches infinity, we can find the value where the population is leveling

We wish to find: First, I will rewrite this as two limits:

The first is a limit of constant so it equals 16. Let’s look at the second limit: This is much like 0/0, it is an indeterminate form so we have to “fix” it with algebra.

The highest degree of t in the problem is 2, so let’s divide top and bottom by t 2 These terms go to 0 as t goes to infinity

Since this limit is 0, the population is approaching the value of 16 as time increases.

Here is an easier way to look at this. Since the degree of the denominator is greater than that of the numerator, it will dominate as t nears infinity. Dividing by a huge number will give a very small value, in fact, it will go to 0. 0