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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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