We have been examining function characteristics near individual numbers. We have noticed that, for the most part, all of the action happens within some interval. Action includes zeros, singularities, hills and valleys in the graph, maximums and minimums. $[-10, 10]$ is a very popular interval to investigate, because of technology. Of course, functions can have zeros outside this interval.

Perhaps we should be investigating the interval $[-1000, 1000]$, $[-50000, 50000]$, $[-750000, 750000]$, or $[-1000000, 1000000]$. Each of our elementary functions has a finite interval which holds all of its interesting places. Even periodic functions have an interval containing a full single wave, which then describes the whole function.

Eventually, outside some interval, the function doesn’t do anything surprising. It settles down into a simple pattern. We call this eventual pattern the end-behavior.

What are the simple patterns on $(-\infty , -a) \cup (a, \infty )$? Where $a$ is “big enough”.

Learning Outcomes

In this section, students will

• examine endbehavior.
• use limit notation.
• examine horizontal asymptotes.