Our functions eventually settle down into simple patterns.

The patterns include

• approach a constant function
• approach a linear function
• continue an oscillating pattern
• approach a basic elementary function

Mostly, we are interested in approaching a constant function. These correspond to horizontal asymptotes on the graph.

Then we are interested in linear patterns (oblique asymptotes)

Then we are interested in if the function becomes unbounded.

Mostly.

But, even more important than these end-behavior discoveries is HOW we decide.

Establishing these end-behaviors often requires a comparison of end-behaviors between functions.

If functions are each tending to infinity, do they do this differently?

If functions are each tending to $0$, do they do this differently?

Learning Outcomes

In this section, students will

• compare endbehaviors of functions.
• establish an order of dominance.