Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.1 Rates of Change and Limits
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 2 What you’ll learn about Average and Instantaneous Speed Definition of Limit Properties of Limits One-Sided and Two-Sided Limits Sandwich Theorem …and why Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 3 Average and Instantaneous Speed
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 4 Definition of Limit
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 5 Definition of Limit continued
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 6 Definition of Limit continued
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 7 Properties of Limits
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 8 Properties of Limits continued Product Rule: Constant Multiple Rule:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 2- 9 Properties of Limits continued
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Properties of Limits
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Polynomial and Rational Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Limits
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Limits
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Limits [-6,6] by [-10,10]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide One-Sided and Two-Sided Limits
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide One-Sided and Two-Sided Limits continued
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example One-Sided and Two-Sided Limits o Find the following limits from the given graph.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Sandwich Theorem
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Sandwich Theorem
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.2 Limits Involving Infinity
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions [-12,12] by [-8,8][-6,6] by [-4,4]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Review Solutions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide What you’ll learn about Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞ …and why Limits can be used to describe the behavior of functions for numbers large in absolute value.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)” we mean the limit of f as x moves increasingly far to the left on the number line.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Horizontal Asymptote
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide [-6,6] by [-5,5] Example Horizontal Asymptote
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Sandwich Theorem Revisited
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Properties of Limits as x→±∞
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Properties of Limits as x→±∞ Product Rule: Constant Multiple Rule:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Properties of Limits as x→±∞
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Infinite Limits as x→a
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Vertical Asymptote
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example Vertical Asymptote [-6,6] by [-6,6]
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide End Behavior Models
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example End Behavior Models
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide End Behavior Models
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide End Behavior Models
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Example “Seeing” Limits as x→±∞
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Quick Quiz Sections 2.1 and 2.2