14.5 Limits Involving Infinity

1. 14.5 Limits Involving Infinity Objective: Define limits involving infinity. Use properties of limits involving infinity. Use the Limit Theorem.

2. Limits Involving Infinity Even though “infinity” (∞) is not a real number, it is convenient to describe a numerical quantity that increases without bound or “negative infinity” (−∞) to describe a numerical quantity decreasing without bound. Earlier we learned how for the graph at right when x approaches 1, 3, or 5 there are no limits, but if we use the infinity symbols it can be useful to describe the behavior of the graph as we approach those values. This is because there are vertical asymptotes at each location.

3. Vertical Asymptotes

4. Example #1 Describe the behavior of the function near x = 0.

5. Example #2 Describe the behavior of the function near x = 3.

6. Limits at Infinity Up until this point every limit we have found involves x approaching some real number c. Now we’ll take a look at what happens when x increases or decreases without bound, or in other words, when x approaches “infinity” or “negative infinity.” This is denoted as follows: The first limit is asking what happens to the function as we go forever to the right. The second limit is asking what happens to the function as we go forever to the left.

7. Limits at Infinity For some functions, as x approaches “infinity” or “negative infinity,” a limit may exist. Take for instance the following function: For both circumstances, the graph will never physically “touch” y = 6 or y = 1, although due to rounding errors it may appear that way on the calculator. These are caused by horizontal asymptotes on the function.

8. Horizontal Asymptotes

9. Example #3 Describe the behavior of the function as x approaches infinity and as x approaches negative infinity.

10. Limits at Infinity In some circumstances as x approaches infinity or negative infinity the limits don’t exist, but the end behavior of the graph can still be described. For the graph at left: In other words, as the graph goes forever right, it goes forever up, and as it goes forever left, it goes forever down.

11. Example #4 Describe the behavior of the function as x approaches infinity and as x approaches negative infinity.

12. Limit Theorem The limit theorem comes from the idea that any fraction with a denominator way larger than its numerator is a very small number. Take for instance the following sequence: and in decimal form: It is clear to see that as the denominator increases, the number gets smaller and smaller and closer and closer to 0. From the negative side, the “smaller” the negative, the greater the number, so for this same sequence, the number would still be approaching 0 if the denominator becomes a larger negative number.

13. Example #5a Describe the end behavior of the function and justify your conclusion.

14. Example #5b Describe the end behavior of the function and justify your conclusion.

15. Example #6a Find each limit.

16. Example #6b Find each limit.

17. Example #6 From the graph of the function you can clearly see two different values are approached for this function as x approaches infinity and as x approaches negative infinity.

18. Example #7 Find each limit.

19. Example #8 Find the horizontal asymptote(s) of the following function. **Hint: Multiply by the conjugate of the denominator. Since there is a definite limit, the horizontal asymptote is y = 1.

20. Example #9 Find the horizontal asymptote(s) of the following function. **Hint: Multiply by the conjugate of the numerator.