1. Lesson 2.2 Limits Involving Infinity  Finite Limits as x->∞  Sandwich Theorem Revisited  Infinite limits as x -> a  End Behavior Models  “Seeing” limits as x ->±∞

2. Asymptotes The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either lim x-> ∞ f(x) = b or lim x-> -∞ f(x) = b Note: A function can have more than one asymptote. The line x = a is a vertical asymptote of the graph of a function y = f(x) if either lim x->a+ f(x) = ± ∞ or lim x-> a- f(x) = ± ∞

3. Example 1 Looking for Horizontal Asymptotes Problem: Use graphs and tables to find lim x -> ∞ f(x), lim x -> - ∞ f(x), and identify all horizontal asymptotes of f(x) = x / √x 2 + 1 GraphicallyNumerically Graph -10 ≤ x ≤ 10. Check table, use ask for large Investigate absolute values to investigate If you believe you found an asymptote, infinity and negative infinity. graph to find overlap. Adjust window as needed Try Exercise 5

4. Sandwich Theorem Revisited! Find lim x -> ∞ f(x) for f(x) = sin x / x. Graphically Check table and graph to see the value of f(x) as x becomes very large (“ ask” with table & large window for domain) Analytically We know that -1 ≤ sin x ≤ 1 To Solve Divide by x, find limits of outside functions. The limit of f(x) must be between them. Try Exercise 12

5. Theorem 5: Properties of Limits! Page 71 If L, M, and k are real numbers then you can 1. Add limits – Sum rule 2. Subtract limits – Difference rule 3. Multiply limits – Product rule 4. Multiply limits by a constant – Constant Multiple Rule 5. Divide limits – Quotient rule (if denominator ≠ 0!) 6. Raise limits to a power, if power ≠ 0! – Power rule

6. Example 3 – Using Theorem 5 Find lim x -> ∞ 5x + sin x x How? Break the function into pieces and find the limit of simpler functions. Then use the sum rule. Try Exercise 25

7. Example 4 Finding Vertical Asymptotes  Find the vertical asymptotes of f(x) = 1 / x 2.  Describe the behavior to the left and right of each vertical asymptote. Look at the table and graph, what can you discern? Try Exercise 28

8. Example 5 Finding Vertical Asymptotes Find the vertical asymptotes for f(x) = tan x = sin x cos x Hint: Where is the function undefined? Try Exercise 31

9. Homework Page 76 Ex. 3-33 (3n, n Є I)

10. Warm Up Teamwork Page 76 Exercises 55, 56, 59-64

11. End Behavior Model The function g is  A right end behavior model for f if and only if lim x -> ∞ f(x) / g(x) = 1.  A left end behavior model for f if and only if lim x -> - ∞ f(x) / g(x) = 1.

12. Example 6 End Behavior Models Let f(x) = 3x 4 – 2x 3 - 3x 2 – 5x + 6 and g(x) = 3x 4. Show that while f and g are quite different for numerically small values of x, they are virtually identical for | x | large. Graphically, use small & large scale Analytically: Try Exercise 40

13. Example 7 Find End Behavior Models for each function f(x) = 2x 5 + x 4 – x 2 + 1 3x 2 – 5x + 7 g(x) = 2x 3 – x 2 + x – 1 5x 3 + x 2 + x – 5 Try Exercise 43

14. Example 8 More End Behavior Let f(x) = x + e -x. Show that g(x) = x is a right end behavior model for f while h(x) = e -x is a left end behavior model for f. Right Left Try Exercise 46

15. Summary  Solve for limits using Substitution Factoring Conjugate Method  Check Graphically Look at large and small windows Find y = b or y = b/0 to determine horizontal and vertical asymptotes  Check Analytically (use a table) Use “ask” with tables to explore large and small values

16. Homework  Page 76 Exercises 35-44, 68  P84 Quick Review 1-10  Study for quiz on 2-1 & 2-2!