hwq. find the following limit: 2 limits at infinity copyright cengage learning. all rights...
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Limits at Infinity Copyright © Cengage Learning. All rights reserved. 3.5TRANSCRIPT
HWQ
5.yat asymptote horizontala and 2,at x asymptote verticala -3,at x hole a
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Limits at Infinity
Copyright © Cengage Learning. All rights reserved.
3.5
4
Determine (finite) limits at infinity.
Determine the horizontal asymptotes, if any, of the graph of a function.
Determine infinite limits at infinity.
Objectives
6
This section discusses the “end behavior” of a function
on an infinite interval. Consider the graph of
as shown in Figure 3.33.
Limits at Infinity
Figure 3.33
7
Graphically, you can see that the values of f(x) appear to approach 3 as x increases without bound or decreases without bound. You can come to the same conclusions numerically, as shown in the table.
Limits at Infinity
8
The table suggests that the value of f(x) approaches 3 as x increases without bound . Similarly, f(x) approaches 3 as x decreases without bound .These limits at infinity are denoted by
and
Limits at Infinity
9
Horizontal Asymptotes
10
In Figure 3.34, the graph of f approaches the line y = L as x increases without bound.
The line y = L is called a horizontal asymptote of the graph of f.
Horizontal Asymptotes
Figure 3.34
1f xx
1lim 0x x
As the denominator gets larger, the value of the fraction gets smaller.
There is a horizontal asymptote if:
limx
f x b
or limx
f x b
Example – Finding a Limit at Infinity
Any constant divided by positive or negative infinity = 0
2lim
1x
x
x 2limx
x
x
This number becomes insignificant as .x
limx
xx
1
There is a horizontal asymptote at 1.
Example – Finding a Limit at Infinity
2lim
1x
x
x 22
lim11
x
x
xx
limx
xx
1
There is a horizontal asymptote at 1.
Same Example – Algebraic Solution
2
lim11
x
x
xx
lim
1 0x
xx
sin xf xx
Example:
sinlimx
xx
Find:
When we graph this function, the limit appears to be zero.
1 sin 1x
so for :0x 1 sin 1xx x x
1 sin 1lim lim limx x x
xx x x
sin0 lim 0x
xx
by the sandwich theorem:
sinlim 0x
xx
Example: 5 sinlimx
x xx
Find:
5 sinlimx
x xx x
sinlim5 limx x
xx
5 0
5
16
Example – Finding a Limit at Infinity
Find the limit:
Solution: Using Theorem 3.10, you can write
17
Example – Finding a Limit at Infinity
Find the limit:
Solution: Note that both the numerator and the denominator approach infinity as x approaches infinity.
18
Example – Solution
This results in an indeterminate form. To resolve this problem, you can divide both the numerator and the denominator by x. After dividing, the limit may be evaluated as shown.
cont’d
19
So, the line y = 2 is a horizontal asymptote to the right.By taking the limit as , you can see that y = 2 is also a horizontal asymptote to the left.
The graph of the function is shown in Figure 3.35. Figure 3.35
Example – Solution cont’d
20
Horizontal Asymptotes
These are the horizontal asymptote rules. Memorize them!
21
Example – Finding a Limit at Infinity
Find the limit:
Solution: 2
5 4 2
3
2 1lim3 5 7x
x x xx x
DNE
Example – Finding a Limit at Infinity
02
3
2 1lim3 5 7x
x xx x
Example – Finding a Limit at Infinity
24
Example – A Function with Two Horizontal Asymptotes
Find each limit.
27
The graph of is shown
in figure 3.38.
Example – Solution
Figure 3.38
cont’d
Often you can just “think through” limits.
1lim sinx x
00
lim sinx
x
0
29
Infinite Limits at Infinity
31
Find each limit.
Solution:
a. As x increases without bound, x3 also increases without bound. So, you can write
b. As x decreases without bound, x3 also decreases without bound. So, you can write
Example 7 – Finding Infinite Limits at Infinity
32
Example 7 – Solution
The graph of f(x) = x3 in Figure 3.42 illustrates these two results. These results agree with the Leading Coefficient Test for polynomial functions.
Figure 3.42
cont’d
Homework
• MMM pgs. 30-31
33
Homework
• Section 3.5• Pg.205, 1-7 odd, 15-33 odd
34
HWQFind the following limit:
35
1
2lim1x
xx
HWQFind the following limit:
36
1lim cosx x
1