We explore functions that behave like horizontal lines as the input grows without bound.

What does the table tell us about as grows bigger and bigger?

Fill in the table below, rounding to decimal places. What does the table tell us about as grows bigger and bigger?

**limit at infinity**of is .

If becomes arbitrarily close to a specific value by making sufficiently large
and negative, we write and we say, the **limit at negative infinity** of is
.

We can assume that all the Limit Laws also apply to limits at infinity.

Sometimes one must be careful, consider this example.

Note, since and we can also apply the Squeeze Theorem when taking limits at infinity. Here is an example of a limit at infinity that uses the Squeeze Theorem, and shows that functions can, in fact, cross their horizontal asymptotes.

It is a common misconception that a function cannot cross an asymptote. As the next example shows, a function can cross a horizontal asymptote, and in the example this occurs an infinite number of times!

We conclude with an infinite limit at infinity.

means that and that .

We see that we may square higher and higher values to obtain larger outputs. This means that is unbounded, and hence .