2. 2.2 Limits Involving Infinity

3. Quick Review In Exercises 1 – 4, find f – 1, and graph f, f – 1, and y = x in the same viewing window.

4. Quick Review In Exercises 1 – 4, find f – 1, and graph f, f – 1, and y = x in the same viewing window.

5. Quick Review In Exercises 5 and 6, find the quotient q (x) and remainder r (x) when f (x) is divided by g(x).

6. Quick Review In Exercises 5 and 6, find the quotient q (x) and remainder r (x) when f (x) is divided by g(x).

7. Quick Review In Exercises 7 – 10, write a formula for (a) f (– x) and (b) f (1/x). Simplify when possible

8. What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞ Essential Question How can limits be used to describe the behavior of functions for numbers large in absolute value?

9. 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.

10. Horizontal Asymptote Example Horizontal Asymptote 1.Use the graph and tables to find each: (c) Identify all horizontal asymptotes.

11. Example Sandwich Theorem Revisited The sandwich theorem also hold for limits as x →  The function oscillates about the x-axis. Therefore y = 0 is the horizontal asymptote.

12. Properties of Limits as x→±∞

13. Product Rule: Constant Multiple Rule: Properties of Limits as x→±∞

15. Infinite Limits as x→a Vertical Asymptote

16. Example Vertical Asymptote 3.Find the vertical asymptotes of the graph of f (x) and describe the behavior of f (x) to the right and left of each vertical asymptote. The vertical asymptotes are: x = – 2 and x = 2 The value of the function approach –  to the left of x = – 2  The value of the function approach +  to the right of x = – 2  The value of the function approach +  to the left of x = 2  The value of the function approach –  to the right of x = 2 

17. End Behavior Models

18. Example End Behavior Models 4.Find the end behavior model for: We can use the end behavior model of a ration function to identify any horizontal asymptote. A rational function always has a simple power function as an end behavior model.

19. Example “Seeing” Limits as x→±∞ We can investigate the graph of y = f (x) as x →  by investigating the graph of y = f (1/x) as x → 0.

20. Pg. 66, 2.1 #1-47 odd