1. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. MCS 122 Chapter 1 Review of Limits Some of the material in these slides is from Calculus 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved.

2. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Two Basic Problems of Calculus The solution to both of these problems requires the use of limits

3. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1.1.1 (p. 70) Limits (An Informal View)‏

4. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.1.9 (p. 71)‏

5. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1.1.2 (p. 72) One-Sided Limits (An Informal View)‏

6. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1.1.3 (p. 73) The Relationship Between One-Sided and Two-Sided Limits

7. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.1.13 (p. 73)‏

8. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.1.3 (p. 77)‏

9. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.1.5 (p. 77)‏

10. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.1.15 (p. 74)‏

11. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1.1.4 (p. 75) Infinite Limits (An Informal View)‏

12. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.1.7 (p. 77)‏

13. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.1.8 (p. 77)‏

14. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.1.9 (p. 78)‏

15. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.2.1 (p. 80)‏ Calculating Limits – Basic Cases

16. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 1.2.1 (p. 80)‏ Calculating Limits – Basic Cases

17. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Calculating Limits – Algebra Rules

18. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.2.2 (p. 87)‏

19. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 1.2.4 (p. 85)‏ Calculating Limits – Rational Functions

20. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1.3.1 (p. 89) Limits at Infinity (An Informal View)‏ Calculating Limits – As x goes to +,- Infinity

21. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.3.2 (p. 89)‏

22. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.3.4 (p. 90)‏

23. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. 1.3.2 (p. 91) Infinite Limits at Infinity (An Informal View)‏ Calculating Limits – As x goes to +,- Infinity

24. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.3.1 (p. 96)‏

25. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.3.2 (p. 96)‏

26. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Exercise 1.3.4 (p. 97)‏

27. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.4.1 (p. 100)‏ Calculating Limits – Rigorous Definition!!

28. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.4.2 (p. 101)‏ Calculating Limits – Rigorous Definition!!

29. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition 1.4.1 (p. 101) Limit Definition Calculating Limits – Rigorous Definition!!

30. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Calculating Limits – Rigorous Definition!! We can calculate that Find a number δ> 0 such that |f(x) – 2| < 0.05 if 0 < |x-1| < δ. (Use the graphing capability of your calculator)

31. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Definition 1.5.1 (p. 110)‏ Continuous Functions

32. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 1.5.3 (p. 113)‏ Continuous Functions

33. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Theorem 1.6.5 (p. 123)‏ Two useful Trig Limits

34. Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Figure 1.6.3 (p. 123)‏ Two useful Trig Limits