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Math 135 Business Calculus Spring 2009

Class Notes

1.1 Limits: A Numerical and Graphical Approach

The limit of a function is the fundamental concept in calculus and is used to define the derivative of afunction, the subject of this first chapter. In this first section, well introduce an intuitive definition ofthe limit of a function.

Calculus is concerned with how function values, or outputs, change as the input changes. Supposewe have a function y = f(x). Then x is the input and y the output. Suppose that, as the input xgets closer and closer to some fixed number a, the function values get closer and closer to some fixednumber L. The number L is called the limit of f as x approaches a.

EXAMPLE Consider the function defined by

f(x) =x2 + x 2

x 1.

First note that f(x) is not defined at x = 1 since the denominator x 1 equals 0 at x = 1. Eventhough the function is not defined at x = 1, we can still examine how the function behaves for x closeto 1. We can do this either numerically by constructing a table of function values for values of x closeto 1, or graphically by looking at its graph near x = 1.

a) Complete the following table of values

x approaches 1 from left | x approaches 1 from right

x 0 0.8 0.9 0.99 0.999 1 1.001 1.01 1.1 1.2 2

f(x)

Based upon the table, what are the function values doing as x gets closer and closer to 1?

b) Use the table of values to sketch the graph of the function close to 1. Based upon the graph, whatare the function values doing as x gets closer and closer to 1?

1 0 1 2 3

x

1

2

3

4

y

1

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2 Chapter 1 Differentiation

In the above example, even though the function is not defined at x = 1,

as x approaches 1 from either side, the function approaches 3.

We use an arrow, , to stand for the words approaches. So the above statement can be written:

As x 1, f(x) 3.

The number 3 is called the limit of f(x) as x approaches 1 from either side and we write

limx1

f(x) = 3.

DEFINITION OF LIMIT

As x approaches a, the limit of f(x) is L, written

limxa

f(x) = L

provided that we can make the values of f(x) arbitrarily close to L (as close to L as we like) by takingx to be sufficiently close to a (on either side of a) but not equal to a.

The notation limxa

f(x) indicates that x approaches a from both the left and the right. If we only

allow x to approach a from the left or from the right, we can consider a one-sided limit. We write

limxa

+f(x) to indicate a limit from the right

andlim

xa

f(x) to indicate a limit from the left

EXAMPLE Consider the piecewise-defined function given by

H(x) =

2x + 2 for x < 12x 4 for x 1

Complete the following table of values and draw a graph of the function to find the limit limx1

H(x), if

it exists. If not, explain why the limit does not exist.

x 0 0.8 0.9 0.99 0.999 1 1.001 1.01 1.1 1.2 2

f(x)

6 5 4 3 2 1 1 2 3 4 5 6 x

654321

1

2

3

4

5

6

y

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1.1 Limits: A Numerical and Graphical Approach 3

In the preceding example, the function H(x) approaches different values as x approaches 1 fromthe left and from the right. The limits from the left and from the right both exist but are not equal toone another. In order for the (two-sided) limit lim

xa

f(x) to exist, both one-sided limits must exist and

be the same.

THEOREM: RELATIONSHIP BETWEEN ONE-SIDED AND TWO-SIDED LIMITS

The limit of f(x), as x approaches a, is L if and only if the limits from the left and the right both

exist and both equal L. That is,limxa

f(x) = L

if and only iflimxa

f(x) = L and limxa

+f(x) = L.

In the first example in this section, the limit limx1

f(x) exists even though the function f(x) is not

defined at x = 1. In the second example, the left-hand limit limx1

H(x) = 4 even though H(1) 6= 4.

THE LIMIT DOES NOT DEPEND UPON THE FUNCTION VALUE

The limit of a function as x approaches a only depends upon the function values close to a and not at

a itself. This means that the limit at a does not depend upon f(a) or even on whether that functionvalue exists.

EXAMPLE Consider the piecewise-defined function given by

G(x) =

5 for x = 2x 1 for x 6= 2

Graph the function and find each of the following limits, if they exist. If necessary state why the limitdoes not exist.

a) limx2

G(x)

b) limx3

G(x) 6 5 4 3 2 1 1 2 3 4 5 6 x

654321

1

2

3

4

5

6

y

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4 Chapter 1 Differentiation

EXAMPLE Consider the function given by

f(x) =1

x 2+ 3

The graph of the function is shown below. Complete the following table of values and use the tableand graph to find the limit lim

x2

f(x), if it exists. If necessary, state why the limit does not exist.

x 1 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 3f(x)

4 3 2 1 1 2 3 4 5 6 7 8x

4

3

2

1

1

2

3

4

5

6

7

8

y

The one-sided limits in the preceding example are and . Keep in mind that and arenot real numbers. We use as a special notation to indicate a quantity which is increasing withoutbound in a positive direction and to indicate a quantity which is increasing without bound in a

negative direction.In addition to having infinite limits as in the preceding example, we can also consider limits at

infinity, denoted bylimx

f(x) or limx

f(x).

EXAMPLE Consider the function f(x) =1

x 2+ 3 in the preceding example. Find lim

x

f(x) and

limx

f(x).

x 10 100 1000 10,000

f(x)

x 10 100 1000 10,000

f(x)