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.

1.1.1 (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 1.1.9 (p. 71)

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)

Calculus

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1.1.3 (p. 73)

The Relationship Between One-Sided and Two-Sided Limits

Calculus

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Figure 1.1.13 (p. 73)

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)

Calculus

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Exercise 1.1.5 (p. 77)

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)

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)

Calculus

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Exercise 1.1.7 (p. 77)

Calculus

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Exercise 1.1.8 (p. 77)

Calculus

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Exercise 1.1.9 (p. 78)

Calculus

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Calculating Limits – Basic Cases

Figure 1.2.1 (p. 80)

Calculus

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Calculating Limits – Basic Cases

Theorem 1.2.1 (p. 80)

Calculus

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Calculating Limits – Algebra Rules

Calculus

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Exercise 1.2.2 (p. 87)

Calculus

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

Calculating Limits – Rational Functions

Theorem 1.2.4 (p. 85)

Calculus

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Calculating Limits – As x goes to +,- Infinity

1.3.1 (p. 89)

Limits at Infinity (An Informal View)

Calculus

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Figure 1.3.2 (p. 89)

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)

Calculus

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

Calculating Limits – As x goes to +,- Infinity

1.3.2 (p. 91)

Infinite Limits at Infinity (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 1.3.1 (p. 96)

Calculus

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Exercise 1.3.2 (p. 96)

Calculus

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Exercise 1.3.4 (p. 97)

Calculus

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Calculating Limits – Rigorous Definition!!

Figure 1.4.1 (p. 100)

Calculus

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Calculating Limits – Rigorous Definition!!

Figure 1.4.2 (p. 101)

Calculus

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Calculating Limits – Rigorous Definition!!

Definition 1.4.1 (p. 101)

Limit Definition

Calculus

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Calculating Limits – Rigorous Definition!!

We can calculate that lim

f f x

1 )

x

 1 3  

x

5 2 Find a number δ> 0 such that |f(x) – 2| < 0.05 if 0 < |x-1| < δ.

Calculus

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

Continuous Functions

Definition 1.5.1 (p. 110)

Calculus

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Continuous Functions

Theorem 1.5.3 (p. 113)

Calculus

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Two useful Trig Limits

Theorem 1.6.5 (p. 123)

Calculus

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

Two useful Trig Limits

Figure 1.6.3 (p. 123)

Calculus

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