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.
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