hwq. find the following limit: 2 limits at infinity copyright cengage learning. all rights...

HWQ

5.yat asymptote horizontala and 2,at x asymptote verticala -3,at x hole a

0,at x zero aith function w a ofequation theWrite

Limits at Infinity

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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