1. Limits involving infinity
Section 2.2
Limits involving infinity

2. Finite Limits as x Approaches Infinity
Definition 1: The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either

3. Horizontal Asymptotes

4. Example 1
Looking for horizontal asymptotes.

5. Example 2
Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.

6. Example 3
Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.

7. Example 4
Use graphs and tables to find (a) and (b). (c) Identify all horizontal asymptotes.

8. Infinite Limits as x Approaches a
Definition 2: The line x = a is a vertical asymptote of the graph of a function y = f(x) if either

9. Vertical Asymptote

10. Example 5
(a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.

11. Example 6
(a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.

12. TOTD Use graphs and tables to find limits.
Use graphs and tables to find the limit as x approaches ±∞. Identify any horizontal asymptotes.
𝑦= 𝑥 𝑥

13. Example 7
(a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.

14. Example 8
(a) Find the vertical asymptotes of the graph of f(x). (b) Describe the behavior of f(x) to the left and right of each vertical asymptote.

15. Example 9

16. Example 10

17. TOTD
Find the vertical asymptotes of the graph of f(x). Describe the behavior of f(x) to the left and right of each vertical asymptote.
𝑓 𝑥 = 𝑥 2 −1 2𝑥+4
Use graphs and tables to find the limit as x approaches ±∞. Identify any horizontal asymptotes.
𝑦= 𝑥𝑠𝑖𝑛𝑥+2𝑠𝑖𝑛𝑥 2 𝑥 2

18. Example 11
Show that while f and g are quite different for numerically small values of x, they are virtually identical for |x| large.

19. End Behavior Models The function is
(a) a right-end behavior model for iff
(b) a left-end behavior model for iff

20. Example 12
(a) Find a power function end behavior model for f. (b) Identify any horizontal asymptotes.

21. Example 13
(a) Find a power function end behavior model for f. (b) Identify any horizontal asymptotes.

22. Example 14
Find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.

23. Example 15
Find (a) a simple basic function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function.

24. Example 16

25. Example 17