OBJECTIVE:

1. DEFINE LIMITS INVOLVING INFINITY.

2. USE PROPERTIES OF LIMITS INVOLVING INFINITY.

3. USE THE LIMIT THEOREM.

14.5 Limits Involving Infinity

Limits Involving Infinity

Even though “infinity” (∞) is not a real number, it is convenient to describe a numerical quantity that increases without bound or “negative infinity” (−∞) to describe a numerical quantity decreasing without bound.

Earlier we learned how for the graph at right when x approaches 1, 3, or 5 there are no limits, but if we use the infinity symbols it can be useful to describe the behavior of the graph as we approach those values. This is because there are vertical asymptotes at each location.

)(lim1

xfx

)(lim3

xfx

)(lim5

xfx

)(lim5

xfx

Vertical Asymptotes

Example #1

Describe the behavior of the function near x = 0.

4

9)(

xxf

)(lim0

xfx

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

Example #2

Describe the behavior of the function near x = 3.

6

5)(

2

xxxg

)(lim

)(lim

3

3

xf

xf

x

x

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

Limits at Infinity

Up until this point every limit we have found involves x approaching some real number c. Now we’ll take a look at what happens when x increases or decreases without bound, or in other words, when x approaches “infinity” or “negative infinity.”

This is denoted as follows:

)(lim xfx

)(lim xfx

The first limit is asking what happens to the function as we go forever to the right.

The second limit is asking what happens to the function as we go forever to the left.

Limits at Infinity

For some functions, as x approaches “infinity” or “negative infinity,” a limit may exist. Take for instance the following function:

1

241

5)(

4

x

e

xf

6)(lim

xfx

1)(lim

xfx

5 10 15 20 25 30 35 40 45 50 55–5–10–15–20–25–30–35–40–45–50–55 x

2

4

6

–2

–4

–6

y

For both circumstances, the graph will never physically “touch” y = 6 or y = 1, although due to rounding errors it may appear that way on the calculator. These are caused by horizontal asymptotes on the function.

Horizontal Asymptotes

Example #3

Describe the behavior of the function as x approaches infinity and as x approaches negative infinity.

3

1)(

xxf

2 4 6 8 10 12–2–4–6–8–10–12 x

2

4

6

8

10

12

–2

–4

–6

–8

–10

–12

y

0)(lim

0)(lim

xf

xf

x

x

Limits at Infinity

In some circumstances as x approaches infinity or negative infinity the limits don’t exist, but the end behavior of the graph can still be described. For the graph at left:

In other words, as the graph goes forever right, it goes forever up, and as it goes forever left, it goes forever down.

)(lim xfx

)(lim xfx

Example #4

Describe the behavior of the function as x approaches infinity and as x approaches negative infinity.

79)( 3 xxxf

2 4 6 8 10 12–2–4–6–8–10–12 x

4

8

12

16

20

24

–4

–8

–12

–16

–20

–24

y

)(lim

)(lim

xf

xf

x

x

Limit Theorem

The limit theorem comes from the idea that any fraction with a denominator way larger than its numerator is a very small number.

Take for instance the following sequence:

and in decimal form:

It is clear to see that as the denominator increases, the number gets smaller and smaller and closer and closer to 0.

From the negative side, the “smaller” the negative, the greater the number, so for this same sequence, the number would still be approaching 0 if the denominator becomes a larger negative number.

,...660.1 0.2, 0.25, ,330. 0.5, 1,

... ,6

1 ,

5

1 ,

4

1 ,

3

1 ,

2

1 ,

1

1

Example #5a

Describe the end behavior of the function and justify your conclusion.

x

xf3

4

5)(

4

5

04

53

4

5)(lim

4

5

04

53

4

5)(lim

xf

xf

x

x

Example #5b

Describe the end behavior of the function and justify your conclusion.

965

524)(

2

2

xx

xxxf

5

496

5

524

965

524

lim965

524

lim)(lim

5

496

5

524

965

524

lim965

524

lim)(lim

2

2

2

2

2

2

2

2

2

2

2

2

xx

xx

x

xxx

xx

xf

xx

xx

x

xxx

xx

xf

xxx

xxx

Example #6a

Find each limit.

35

12lim

2

x

x

x

5

23

5

12

35

12

lim

35

12

lim35

12

lim35

12

lim

2

2

2

2

22

x

x

x

xx

x

x

xx

x

x

xx

x

x

xxx

0: 2 xforxxNote

Example #6b

Find each limit.

35

12lim

2

x

xx

0: 22 xforxxxxNote

5

23

5

12

35

12

lim

35

12

lim35

12

lim35

12

lim

2

2

2

2

22

x

x

xx

x

x

xx

x

x

xxx

x

x

xxx

Example #6

From the graph of the function you can clearly see two different values are approached for this function as x approaches infinity and as x approaches negative infinity.

35

12)(

2

x

xxf

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

Example #7

Find each limit.

12

434

22

45lim

2

2

xx

xx

x

xx

2

1

2

4

2

5

102

004

02

0511

2

434

22

45

112

434

22

45

lim12

434

22

45

lim

2

2

2

2

2

2

xx

xx

x

x

x

xxx

xx

x

xx

x

xx

Example #8

Find the horizontal asymptote(s) of the following function.

7

5)(

x

xxf

11

1

01

00149

1

35121

491

35121

lim49

3512

lim

49

3512lim

49

3557lim

7

7

7

5lim

7

5lim

x

xx

xx

xxx

x

xx

x

xxx

x

x

x

x

x

x

xx

xxxx

Since there is a definite limit, the horizontal asymptote is y = 1.

**Hint: Multiply by the conjugate of the denominator.

Example #9

Find the horizontal asymptote(s) of the following function.

xxxf 2)( 2

01

0

1

0

001

0

421

2

421

2

lim42

2

lim42

2

lim42

2lim

42

2lim

2

2

1

2lim2lim

22

222

2

22

2

222

xx

x

xx

x

x

x

x

x

x

x

xx

xx

xxxxxx

xxxx

xxx

**Hint: Multiply by the conjugate of the numerator.