1. HWQ

2. Find the following limit: 2

3. Limits at Infinity Copyright © Cengage Learning. All rights reserved. 3.5

4. 4 Determine (finite) limits at infinity. Determine the horizontal asymptotes, if any, of the graph of a function. Determine infinite limits at infinity. Objectives

5. 5 Limits at Infinity

6. 6 This section discusses the “end behavior” of a function on an infinite interval. Consider the graph of as shown in Figure 3.33. Limits at Infinity Figure 3.33

7. 7 Graphically, you can see that the values of f(x) appear to approach 3 as x increases without bound or decreases without bound. You can come to the same conclusions numerically, as shown in the table. Limits at Infinity

8. 8 The table suggests that the value of f(x) approaches 3 as x increases without bound. Similarly, f(x) approaches 3 as x decreases without bound. These limits at infinity are denoted by and Limits at Infinity

9. 9 Horizontal Asymptotes

10. 10 In Figure 3.34, the graph of f approaches the line y = L as x increases without bound. The line y = L is called a horizontal asymptote of the graph of f. Horizontal Asymptotes Figure 3.34

11. As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or Example – Finding a Limit at Infinity Any constant divided by positive or negative infinity = 0

12. This number becomes insignificant as. There is a horizontal asymptote at 1. Example – Finding a Limit at Infinity

13. There is a horizontal asymptote at 1. Same Example – Algebraic Solution

14. Example: Find: When we graph this function, the limit appears to be zero. so for : by the sandwich theorem:

15. Example: Find:

16. 16 Example – Finding a Limit at Infinity Find the limit: Solution: Using Theorem 3.10, you can write

17. 17 Example – Finding a Limit at Infinity Find the limit: Solution: Note that both the numerator and the denominator approach infinity as x approaches infinity.

18. 18 Example – Solution This results in an indeterminate form. To resolve this problem, you can divide both the numerator and the denominator by x. After dividing, the limit may be evaluated as shown. cont’d

19. 19 So, the line y = 2 is a horizontal asymptote to the right. By taking the limit as, you can see that y = 2 is also a horizontal asymptote to the left. The graph of the function is shown in Figure 3.35. Figure 3.35 Example – Solution cont’d

20. 20 Horizontal Asymptotes These are the horizontal asymptote rules. Memorize them!

21. 21 Example – Finding a Limit at Infinity Find the limit: Solution: 2

22. Example – Finding a Limit at Infinity

24. 24 Example – A Function with Two Horizontal Asymptotes Find each limit.

25. 25 For x > 0, you can write. So, dividing both the numerator and the denominator by x produces and you can take the limits as follows. Example 4(a) – Solution

26. 26 For x < 0, you can write So, dividing both the numerator and the denominator by x produces and you can take the limits as follows. Example 4(b) – Solution cont’d

27. 27 The graph of is shown in figure 3.38. Example – Solution Figure 3.38 cont’d

28. Often you can just “think through” limits. 

29. 29 Infinite Limits at Infinity

30. 30 Infinite Limits at Infinity

31. 31 Find each limit. Solution: a.As x increases without bound, x 3 also increases without bound. So, you can write b.As x decreases without bound, x 3 also decreases without bound. So, you can write Example 7 – Finding Infinite Limits at Infinity

32. 32 Example 7 – Solution The graph of f(x) = x 3 in Figure 3.42 illustrates these two results. These results agree with the Leading Coefficient Test for polynomial functions. Figure 3.42 cont’d

33. Homework MMM pgs. 30-31 33

34. Homework Section 3.5 Pg.205, 1-7 odd, 15-33 odd 34

35. HWQ Find the following limit: 35

36. HWQ Find the following limit: 36