Connecting Limits at Infinity and Horizontal Asymptotes

What Limits at Infinity Tell You

A limit at infinity describes the end behavior of a function. Instead of asking what happens as approaches a particular number, we ask what happens to as becomes arbitrarily large in the positive or negative direction.

For example, consider . As , the denominator becomes much larger than the constant numerator, so the fraction gets closer and closer to zero.

The important idea is that the function does not have to reach zero. The limit describes what value the function approaches as moves farther and farther to the right.

From a Limit to a Horizontal Asymptote

A horizontal asymptote describes the horizontal line that a function approaches as or . The connection is direct:

So if you calculate a limit at infinity and obtain , the graph has a horizontal asymptote at .

Likewise, if the limit as is , then the graph approaches on its left-hand end.

Right-End and Left-End Behavior

There are actually two separate limits at infinity to consider:

These two limits can be different. A function can therefore approach one horizontal asymptote as and a different horizontal asymptote as .

For example, the function approaches on the right but on the left. This gives the graph two different horizontal asymptotes.

How to Find Limits at Infinity for Rational Functions

For rational functions, the easiest method is usually to compare the degrees of the numerator and denominator.

1. Numerator degree less than denominator degree: the limit is .
2. Numerator degree equal to denominator degree: the limit is the ratio of the leading coefficients.
3. Numerator degree greater than denominator degree: the limit does not approach a finite horizontal value, so there is no horizontal asymptote from that end.

Reading a Graph

Sometimes the AP exam gives you a graph instead of an equation. In that case, you can determine the limit by looking at the function's end behavior.

• If the right-hand end of the graph approaches , then .
• If the left-hand end approaches , then .
• If both ends approach the same value , then the function has the horizontal asymptote .

Be careful: a horizontal asymptote describes end behavior. The graph can cross a horizontal asymptote, and it can even cross the same horizontal asymptote many times.

Horizontal Asymptotes Do Not Mean the Function Cannot Reach the Value

A common misconception is that a function can never equal its horizontal asymptote. That's not true. An asymptote describes what happens as becomes arbitrarily large in magnitude; it does not prohibit the function from crossing or touching that line at finite values of .

Common Mistakes

Practice Problems