limits 2. 2.2 the limit of a function limits in this section, we will learn: about limits in general...

LIMITSLIMITS

2

2.2The Limit of a Function

LIMITS

In this section, we will learn:

About limits in general and about numerical

and graphical methods for computing them.

Let’s investigate the behavior of the

function f defined by f(x) = x2 – x + 2

for values of x near 2. The following table gives values of f(x) for values of x

close to 2, but not equal to 2.

THE LIMIT OF A FUNCTION

p. 66

From the table and the

graph of f (a parabola)

shown in the figure,

we see that, when x is

close to 2 (on either

side of 2), f(x) is close

to 4.

THE LIMIT OF A FUNCTION

Figure 2.2.1, p. 66

We express this by saying “the limit of

the function f(x) = x2 – x + 2 as x

approaches 2 is equal to 4.” The notation for this is:

2

2lim 2 4x

x x

THE LIMIT OF A FUNCTION

In general, we use the following

notation. We write

and say “the limit of f(x), as x approaches a, equals L”

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

limx a

f x L

THE LIMIT OF A FUNCTION Definition 1

An alternative notation for

is as

which is usually read “f(x) approaches L as

x approaches a.”

limx a

f x L

THE LIMIT OF A FUNCTION

( )f x L x a

Notice the phrase “but x a” in the

definition of limit.

THE LIMIT OF A FUNCTION

21

1lim

1x

x

x

THE LIMIT OF A FUNCTION Example 1

lim ( )x a

f x

Guess the value of .

Notice that the function f(x) = (x – 1)/(x2 – 1) is not defined when x = 1.

However, that doesn’t matter—because the definition of says that we consider values of x that are close to a but not equal to a.

The tables give values

of f(x) for values of x that

approach 1 (but are not

equal to 1). On the basis of the values,

we make the guess that

Solution: Example 1

21

1lim 0.5

1x

xx

p. 67

Example 1 is illustrated by the graph

of f in the figure.

THE LIMIT OF A FUNCTION Example 1

Figure 2.2.3, p. 67

Now, let’s change f slightly by giving it thevalue 2 when x = 1 and calling the resultingfunction g:

This new function g stillhas the same limit as x approaches 1.

2

11

12 1

xif x

g x xif x

THE LIMIT OF A FUNCTION Example 1’

Estimate the value of .

The table lists values of the function for several values of t near 0.

As t approaches 0, the values of the function seem to approach 0.16666666…

So, we guess that:

2

20

9 3limt

t

t

THE LIMIT OF A FUNCTION Example 2

2

20

9 3 1lim

6t

t

t

p. 68

These figures show quite accurate graphs

of the given function, we can estimate easily

that the limit is about 1/6.

THE LIMIT OF A FUNCTION Example 2

Figure 2.2.5, p. 68

However, if we zoom in too much, then

we get inaccurate graphs—again because

of problems with subtraction.

THE LIMIT OF A FUNCTION Example 2

Figure 2.2.5, p. 68

Guess the value of .

The function f(x) = (sin x)/x is not defined when x = 0. Using a calculator (and remembering that, if ,

sin x means the sine of the angle whose radian measure is x), we construct a table of values correct to eight decimal places.

0

sinlimx

x

x

THE LIMIT OF A FUNCTION Example 3

x

p. 69

From the table and the graph, we guess that

This guess is, in fact, correct—as will be proved in Chapter 3, using a geometric argument.

0

sinlim 1x

x

x

Solution: Example 3

p. 69Figure 2.2.6, p. 69

Investigate .

Again, the function of f(x) = sin ( /x) is undefined at 0.

0limsinx x

THE LIMIT OF A FUNCTION Example 4

Evaluating the function for some small values of x, we get:

Similarly, f(0.001) = f(0.0001) = 0.

THE LIMIT OF A FUNCTION Example 4

1 sin 0f 1sin 2 0

2f

1sin 3 0

3f

1sin 4 0

4f

0.1 sin10 0f 0.01 sin100 0f

On the basis of this information,

we might be tempted to guess

that .

This time, however, our guess is wrong. Although f(1/n) = sin n = 0 for any integer n, it is

also true that f(x) = 1 for infinitely many values of x that approach 0.

0limsin 0x x

THE LIMIT OF A FUNCTION Example 4

Wrong

The graph of f is given in the figure. the values of sin( /x) oscillate between 1 and –1

infinitely as x approaches 0. Since the values of f(x) do not approach a fixed

number as approaches 0, does not exist.

Solution: Example 4

Figure 2.2.7, p. 69

0limsinx x

Find .

As before, we construct a table of values. From the table, it appears that:

Later, we will see that:

3

0

cos5lim 0

10,000x

xx

3

0

cos5lim

10,000x

xx

THE LIMIT OF A FUNCTION Example 5

p. 70

0lim cos5 1x x

Wrong

Examples 4 and 5 illustrate some of the

pitfalls in guessing the value of a limit. It is easy to guess the wrong value if we use

inappropriate values of x, but it is difficult to know when to stop calculating values.

As the discussion after Example 2 shows, sometimes, calculators and computers give the wrong values.

In the next section, however, we will develop foolproof methods for calculating limits.

THE LIMIT OF A FUNCTION

The Heaviside function H is defined by:

The function is named after the electrical engineer Oliver Heaviside (1850–1925).

It can be used to describe an electric current that is switched on at time t = 0.

0 1

1 0

if tH t

if t

THE LIMIT OF A FUNCTION Example 6

The graph of the function is shown in

the figure. As t approaches 0 from the left, H(t) approaches 0. As t approaches 0 from the right, H(t) approaches 1. There is no single number that H(t) approaches as t

approaches 0. So, does not exist.

Solution: Example 6

0limt H t

Figure 2.2.8, p. 70

0lim 0t

H t

0lim 1t

H t

We write

and say the left-hand limit of f(x) as x

approaches a—or the limit of f(x) as x

approaches a from the left—is equal to L if

we can make the values of f(x) arbitrarily

close to L by taking x to be sufficiently close

to a and x less than a.

limx a

f x L

ONE-SIDED LIMITS Definition 2

ONE-SIDED LIMITS

The definitions are illustrated in the

figures.

Figure 2.2.9, p. 71

By comparing Definition 1 with the definition

of one-sided limits, we see that the following

is true:

lim lim limx a x a x a

f x L if and only if f x L and f x L

ONE-SIDED LIMITS

The graph of a function g is displayed. Use it

to state the values (if they exist) of:

2

limx

g x

2

limx

g x

2

limxg x

5limx

g x

5

limx

g x

5

limxg x

ONE-SIDED LIMITS Example 7

Figure 2.2.10, p. 71

(a) and

=> does NOT exist.

(b) and

=>

■ notice that .

2

lim 3x

g x

2

lim 1x

g x

Solution: Example 7

Figure 2.2.10, p. 71

2

limxg x

5

lim 2x

g x

5

lim 2x

g x

5

lim 2xg x

5 2g

Find if it exists.

As x becomes close to 0, x2 also becomes close to 0, and 1/x2 becomes very large.

20

1limx x

INFINITE LIMITS Example 8

p. 72

To indicate the kind of behavior exhibited

in the example, we use the following

notation:

This does not mean that we are regarding ∞ as a number. Nor does it mean that the limit exists. It simply expresses the particular way in which the limit

does not exist. 1/x2 can be made as large as we like by taking x close

enough to 0.

0 2

1limx x

Solution: Example 8

Let f be a function defined on both sides

of a, except possibly at a itself. Then,

means that the values of f(x) can be

made arbitrarily large—as large as we

please—by taking x sufficiently close to a,

but not equal to a.

limx a

f x

INFINITE LIMITS Definition 4

Another notation for is: limx a

f x

INFINITE LIMITS

f x as x a

Let f be defined on both sides of a, except

possibly at a itself. Then,

means that the values of f(x) can be made

arbitrarily large negative by taking x

sufficiently close to a, but not equal to a.

limx a

f x

INFINITE LIMITS Definition 5

The symbol can be read

as ‘the limit of f(x), as x approaches a,

is negative infinity’ or ‘f(x) decreases

without bound as x approaches a.’ As an example,

we have:

20

1limx x

INFINITE LIMITS

limx a

f x

Similar definitions can be given for the

one-sided limits:

Remember, ‘ ’ means that we consider only values of x that are less than a.

Similarly, ‘ ’ means that we consider only .

limx a

f x

limx a

f x

limx a

f x

limx a

f x

INFINITE LIMITS

x a

x a x a

Those four cases are illustrated here.

INFINITE LIMITS

Figure 2.2.14, p. 73

The line x = a is called a vertical asymptote

of the curve y = f(x) if at least one of the

following statements is true.

For instance, the y-axis is a vertical asymptote of the curve y = 1/x2 because .

limx a

f x

limx a

f x

limx a

f x

limx a

f x

limx a

f x

limx a

f x

INFINITE LIMITS Definition 6

0 2

1limx x

In the figures, the line x = a is a vertical

asymptote in each of the four cases shown. In general, knowledge of vertical asymptotes is very

useful in sketching graphs.

INFINITE LIMITS

Figure 2.2.14, p. 73

Find and .

If x is close to 3 but larger than 3, then the denominator x – 3 is a small positive number and 2x is close to 6.

So, the quotient 2x/(x – 3) is a large positive number.

Thus, intuitively, we see that .

3

2lim

3x

x

x 3

2lim

3x

x

x

INFINITE LIMITS Example 9

3

2lim

3x

x

x

The graph of the curve y = 2x/(x - 3) is

given in the figure. The line x – 3 is a vertical asymptote.

Solution: Example 9

Figure 2.2.15, p. 74

3

2lim

3x

x

x

3

2lim

3x

x

x

infinity

Nagative infinity

Find the vertical asymptotes of

f(x) = tan x. As , there are potential vertical

asymptotes where cos x = 0. In fact, since as and

as , whereas sin x is positive when x is near /2, we have:

and

This shows that the line x = /2 is a vertical asymptote.

INFINITE LIMITS Example 10

sintan

cos

xx

x

cos 0x / 2x cos 0x / 2x

/ 2lim tan

xx

/ 2lim tan

xx

Similar reasoning shows that the

lines x = (2n + 1) /2, where n is an

integer, are all vertical asymptotes of

f(x) = tan x. The graph confirms this.

Solution: Example 10

Figure 2.2.16, p. 74