1. Slide 1.1- 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Limits: A Numerical and Graphical Approach OBJECTIVE  Find limits of functions, if they exist, using numerical or graphical methods. 1.1

3. Slide 1.1- 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley DEFINITION: As x approaches a, the limit of f (x) is L, written if all values of f (x) are close to L for values of x that are sufficiently close, but not equal to, a. 1.1 Limits: A Numerical and Graphical Approach

4. Slide 1.1- 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM: As x approaches a, the limit of f (x) is L, if the limit from the left exists and the limit from the right exists and both limits are L. That is, if 1) and 2) then 1.1 Limits: A Numerical and Graphical Approach

5. Slide 1.1- 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: Consider the function H given by Graph the function, and find each of the following limits, if they exist. When necessary, state that the limit does not exist. a) b) 1.1 Limits: A Numerical and Graphical Approach

6. Slide 1.1- 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) Limit Numerically First, let x approach 1 from the left: Thus, it appears that 00.50.80.90.990.999 H(x)H(x)233.63.83.983.998 1.1 Limits: A Numerical and Graphical Approach

7. Slide 1.1- 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) Limit Numerically (continued) Then, let x approach 1 from the right: Thus, it appears that 21.81.11.011.0011.0001 H(x)H(x)0–0.4–1.8–1.98–1.998–1.9998 1.1 Limits: A Numerical and Graphical Approach

8. Slide 1.1- 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) Limit Numerically (concluded) Since1) and 2) Then, does not exist. 1.1 Limits: A Numerical and Graphical Approach

9. Slide 1.1- 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) Limit Graphically Observe on the graph that: 1) and 2) Therefore, does not exist. 1.1 Limits: A Numerical and Graphical Approach

10. Slide 1.1- 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley b) Limit Numerically First, let x approach –3 from the left: Thus, it appears that –4–3.5–3.1–3.01–3.001 H(x)H(x)–6–5–4.2–4.02–4.002 1.1 Limits: A Numerical and Graphical Approach

11. Slide 1.1- 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley b) Limit Numerically (continued) Then, let x approach –3 from the right: Thus, it appears that –2–2.5–2.9–2.99–2.999 H(x)H(x)–2–3–3.8–3.98–3.998 1.1 Limits: A Numerical and Graphical Approach

12. Slide 1.1- 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley b) Limit Numerically (concluded) Since1) and 2) Then, 1.1 Limits: A Numerical and Graphical Approach

13. Slide 1.1- 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley b) Limit Graphically Observe on the graph that: 1) and 2) Therefore, 1.1 Limits: A Numerical and Graphical Approach

14. Slide 1.1- 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The “Wall” Method: As an alternative approach to Example 1, we can draw a “wall” at x = 1, as shown in blue on the following graphs. We then follow the curve from left to right with pencil until we hit the wall and mark the location with an ×, assuming it can be determined. Then we follow the curve from right to left until we hit the wall and mark that location with an ×. If the locations are the same, we have a limit. Otherwise, the limit does not exist. 1.1 Limits: A Numerical and Graphical Approach

15. Slide 1.1- 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Thus for Example 1: does not exist 1.1 Limits: A Numerical and Graphical Approach

16. Slide 1.1- 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3: Consider the function f given by Graph the function, and find each of the following limits, if they exist. If necessary, state that the limit does not exist. a) b) 1.1 Limits: A Numerical and Graphical Approach

17. Slide 1.1- 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) Limit Numerically Let x approach 3 from the left and right: Thus, 2.12.52.92.99 f (x)135 3.53.23.13.01 f (x) 1.1 Limits: A Numerical and Graphical Approach

18. Slide 1.1- 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) Limit Graphically Observe on the graph that: 1) and 2) Therefore, 1.1 Limits: A Numerical and Graphical Approach

19. Slide 1.1- 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley b) Limit Numerically Let x approach 2 from the left and right: Thus, does not exist. 1.51.91.991.999 f (x)1–7–97–997 2.52.12.012.001 f (x)5131031003 1.1 Limits: A Numerical and Graphical Approach

20. Slide 1.1- 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley b) Limit Graphically Observe on the graph that: 1) and 2) Therefore, does not exist. 1.1 Limits: A Numerical and Graphical Approach

21. Slide 1.1- 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4: Consider again the function f given by Find 1.1 Limits: A Numerical and Graphical Approach

22. Slide 1.1- 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Limit Numerically Note that you can only approach ∞ from the left: Thus, 5101001000 f (x)3.1253.01023.001 1.1 Limits: A Numerical and Graphical Approach

23. Slide 1.1- 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Limit Graphically Observe on the graph that, again, you can only approach ∞ from the left. Therefore, 1.1 Limits: A Numerical and Graphical Approach