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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

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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

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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

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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

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