Limits Involving Infinity Chapter 2: Limits and Continuity

What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models …and why Limits can be used to describe the behavior of functions for numbers large in absolute value.

Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line.

Example 1 - Horizontal Asymptote By looking at the graph and a table of values, it appears that as we head off in both the positive and negative x directions towards infinity, that the graph gets closer and closer to 1. Thinking about the function f(x), it makes sense that the function will never actually equal 1 in either direction as we head to infinity but will just continue to get infinitely closer to 1. Using the graph, the table, and our intuition about the given function f(x), we can give the following answers: Note that we couldn’t just use direct substitution here. Plugging in ∞ for x doesn’t make any sense.

Example 2 - Sandwich Theorem Revisited A lot of students don’t like this problem because they learned that a function can never cross a horizontal asymptote. Just think of an asymptote as something that the function gets closer to as we head to ±∞, don’t worry about the function crossing it.

Properties of Limits as x→±∞

Constant Multiple Rule: Product Rule:

Properties of Limits as x→±∞ These properties are essentially the same (except they apply as x → ∞) as the original properties that we learned for limits as x → c. You will do examples in your homework that will require you to use these rules.

End Behavior Models This is a pretty technical definition and we won’t really use it in practice much but I will demonstrate it in an example in an upcoming slide. The purpose of an end behavior model is really just to help us visualize what is happening as the function heads off to ±∞.

Example 3 - End Behavior Models In general, when we are looking at end behavior models of a rational function, just circle the term with the highest power of x in the numerator and the denominator. Simplify those terms with respect to each other and that will be your end behavior model. This will also help find any horizontal asymptotes that may exist.

Example 4 - End Behavior Models

Infinite Limits as x→a In the previous sections, we had said that when this occurred, that the limit D.N.E.(does not exist). We are now kind of replacing that when both the left and right limits approach the same ±∞ from both sides. For AP exam purposes, saying D.N.E or ±∞ are often interchangeable. An exception to this is when one of the one sided limits goes to positive ∞ and the other goes to negative ∞. In that case there, we would still need to say that that the limit D. N. E.

Example 5 - Vertical Asymptote Let’s take a look at the graph of the function.

Summary