Ch11 Limits Apreviewtocalculus

  • Published on
    20-Jul-2016

  • View
    25

  • Download
    0

Embed Size (px)

DESCRIPTION

Preview to Limits

Transcript

  • Limits: A Preview to Calculus

    A112-acre parcel of landin Pavo, Georgia, isbounded by a street,

    two other property lines,

    and Little Creek, which

    runs along the back of it.

    How did the surveyors in Brooks County determine that the parcel is 112 acres? In this chapter, you will

    use limits to nd the area of a region with a curved boundary. If we consider Little Creek a function, then

    the area below the curve of the function, above Old Pavo Road, and between the two side property

    lines represents the parcel of land. Surveyors use limits (which are fundamental later in calculus) to

    determine areas of irregular parcels.*

    11

    *See Section 11.5, Exercises 41 and 42.

    OldOld PavoPavo RoadRoadOld Pavo Road

    Little Creek

    c11aLimitsAPreviewtoCalculus.qxd 6/10/13 3:55 PM Page 1076

  • 1077

    IN THIS CHAPTER we will rst dene what a limit of a function is and then discuss how to nd limits of functions. We will discuss nding limits numerically with tables, graphically, and algebraically. We will use limits to dene tangentlines to curves and then to dene the slope of a functionthe derivative. We will discuss limits at innity and the limits ofsequences and summations, and then apply limits to application problems like nding the area below a curve.

    Limit Laws Finding Limits

    Using Limit Laws Finding Limits

    Using DirectSubstitution

    Finding LimitsUsing AlgebraicTechniques

    Finding LimitsUsing Left-Handand Right-HandLimits

    Denition ofa Limit

    Estimating LimitsNumerically andGraphically

    Limits That Failto Exist

    One-Sided Limits

    Tangent Lines The Derivative

    of a Function Instantaneous

    Rates of Change

    Limits at Innity Limits of

    Sequences

    Limits ofSummations

    The Area Problem

    LIMITS: A PREVIEW TO CALCULUS

    L E A R N I N G O B J E C T I V E S

    Understand the meaning of a limit and be able to estimate limits. Apply limit laws and algebraic techniques to nd exact values of limits and understand how

    these techniques differ from estimating techniques. Find the tangent line to an arbitrary point on a curve representing a function

    and understand how the slope of that line corresponds to the derivative of that function. Find limits at innity. Use the limits of summations to nd the area under a curve.

    11.1Introduction to

    Limits:Estimating LimitsNumerically and

    Graphically

    11.2Techniques forFinding Limits

    11.3Tangent Lines

    and Derivatives

    11.4Limits at Innity;

    Limits ofSequences

    11.5Finding the Area

    Under a Curve

    c11aLimitsAPreviewtoCalculus.qxd 6/10/13 3:55 PM Page 1077

  • 1078

    Definition of a Limit

    The notion of a limit is a fundamental concept in calculus. The question: What happensto the values f (x) of a function f as x approaches the real number a? can be answeredwith a limit.

    Let us consider the quadratic function If we rewrite this functionas we can see its graph is a parabola opening upward with a vertex at thepoint (2, 0). Lets investigate the behavior of this function f as x approaches 3. We start bylisting a table of values of f(x) for values of x near 3, but not equal to 3. It is important totake values of x approaching from both the left (values less than 3) and the right (valuesgreater than 3).

    f (x) = (x - 2)2,f (x) = x2 - 4x + 4.

    y

    1 2 3 4x

    1

    2

    3

    4

    As x approaches 3

    f (x)approaches

    1

    f (x) = (x 2)2x 2.9 2.99 2.999 3 3.001 3.01 3.1

    f(x) 0.810 0.980 0.998 ? 1.002 1.020 1.21

    x approaches 3 from the left x approaches 3 from the right

    f(x) approaches 1 f (x) approaches 1

    We see in both the table and the graph that when x is close to 3, the values of f (x) areclose to 1. In other words, as x approaches 3 from either side (left or right), the values off(x) approach 1.

    WORDS MATH

    The limit of the function as x

    approaches 3 is equal to 1.

    You may be thinking that if we had evaluated the function at we wouldhave found it to be equal to 1. Although that is true, the concept of a limit is the behaviorof the function as x approaches a value. In fact, a function does not even have to be definedat a value for a limit to exist at that value.

    f (3) = 1,x = 3,

    f (x) = x2 - 4x + 4limxS3

    (x2 - 4x + 4) = 1

    CONCEPTUAL OBJECTIVES

    Understand that the limit of a function at a point mayexist even though the function may not be defined atthat point.

    Understand when a limit of a function fails to exist. Understand the difference between a limit of a function

    and a one-sided limit of a function.

    INTRODUCTION TO LIMITS: ESTIMATINGLIMITS NUMERICALLY AND GRAPHICALLY

    SKILLS OBJECTIVES

    Use tables of values to estimate limits of functions numerically.

    Estimate limits of functions by inspecting graphs. Determine whether limits of functions exist.

    S E C T I O N

    11.1

    c11aLimitsAPreviewtoCalculus.qxd 6/10/13 3:55 PM Page 1078

  • 11.1 Introduction to Limits: Estimating Limits Numerically and Graphically 1079

    In other words, the values of f(x) keep getting closer and closer to some real number L as xkeeps getting closer and closer to some real number a (from either side of a). An alternativenotation for is

    as

    This is read as f(x) approaches L as x approaches a. This is the notation we used inSection 2.6 when discussing asymptotes of rational functions.

    It is important to note in defining the limit as that we do not set regardless of whether x can equal a. This means that we are interested only in the valuesof x close to a and we do not even consider when For all three figures below,

    even though in part (b), and in part (c), f(a) is not defined.f (a) Z L,limxSa

    f (x) = Lx = a.

    x = a,x S a

    x S af (x) S L

    limxSa

    f (x) = L

    If the values of f(x) become arbitrarily close to L as x gets sufficiently close to a,but not equal to a, then

    WORDS MATH

    The limit of f(x), as x approaches a, is L. limxSa

    f (x) = L

    The Limit of a FunctionDE F I N I T I O N Study Tip

    Imagine a very small differencebetween x and a, and imaginemaking it smaller and smaller. In thesame way, the difference betweenf(x) and L gets smaller and smaller.

    y

    a

    L

    x

    y

    a

    L

    x

    y

    a

    L

    x

    f(a) is not definedf (a) Z Lf (a) = L

    limxSa

    f (x) = LlimxSa

    f (x) = LlimxSa

    f (x) = L

    (a) (b) (c)

    Estimating Limits Numerically and Graphically

    In this section, we use calculators to make tables of values of functions and we inspectgraphs of functions to surmise whether a limit of a function exists and, if so, to estimatelimits of functions. It is important to note now (and we will summarize again at the end ofthis section) that it is possible for calculators and graphing technologies to give incorrectvalues and pictures of behaviors. In the next section, however, we will discuss analyticmethods for calculating limits, which are foolproof.

    c11aLimitsAPreviewtoCalculus.qxd 6/10/13 3:55 PM Page 1079

  • x 0.9 0.99 0.999 1 1.001 1.01 1.1

    f(x) 1.9 1.99 1.999 ? 2.001 2.01 2.1

    x approaches 1 from the left x approaches 1 from the right

    f (x) approaches 2 f (x) approaches 2

    EXAMPLE 1 Estimating a Limit Numerically and Graphically

    Estimate the value of using a table of values and a graph.

    Solution:

    STEP 1 Make a table with values of x approaching 1 from both the left and the right.

    limxS1

    x2 - 1x - 1

    Technology Tip

    Both the table and the graph indicatethat f(x) approaches 2 as xapproaches 1.

    STEP 2 Draw the graph of

    and inspect the behavior of f (x) as x approaches 1 from both the left and the right.

    f (x) =x2 - 1x - 1

    Both the table and the graph indicate that our estimate should be 2. limxS1

    x2 - 1x - 1

    = 2

    Answer: -4

    Study Tip

    Notice in Example 1 that the limit of

    as exists even

    though is not in the domain of f.x = 1

    x S 1f (x) =x2 - 1x - 1

    y

    1 2 3 4x

    1

    2

    3

    4

    As x approaches 1

    f (x)approaches

    2

    f (x) = x2 1

    x 1

    YOUR TURN Estimate the value of using a table of values and a graph.limxS-2

    x2 - 4x + 2

    The graph shows that is not inthe domain of f.

    x = 1

    1080 CHAPTER 11 Limits: A Preview to Calculus

    Classroom Example 11.1.1 Answer:Estimate the following limits using a table of values. a.10 b.

    a. b.* limxS14

    x3 - 116 x

    x - 14limxS5

    25 - x2

    x - 5

    18

    c11aLimitsAPreviewtoCalculus.qxd 6/10/13 3:55 PM Page 1080

  • Technology Tip

    Set the viewing rectangle asby

    Both the table and thegraph indicate that f(x) approaches 0 as x approaches 0.

    [-0.3, 0.4].[-0.000021, 0.000021]

    Study Tip

    The notation corresponds toand is often used when

    zooming in on graphs when valuesare very small.

    2 * 10-82E - 8

    EXAMPLE 2 Tables and Graphing Technology PitfallsWhen Estimating Limits

    Estimate the value of using a table of values and a graphing utility.

    Solution:

    If we zoom in closer and closer (x approaches 0 from both sides), one might be led to believe from the table and the graph that the limit is equal to 0.

    limxS0

    2x2 + 4 - 2

    x2

    x 0 0.00000001 0.0000001 0.00001

    f(x) 0.25000 0.26645 0.00000 ? 0.00000 0.26645 0.25000

    -0.00000001-0.0000001-0.00001

    1E-6 6E-7 2E-7 2E-7 6E-7 1E-60.020.06

    x

    y0.38

    0.1

    0.340.3

    0.260.220.180.14

    1E-7 6E-8 2E-8 2E