LIMITS AND DERIVATIVES 2. In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The.

LIMITS AND DERIVATIVESLIMITS AND DERIVATIVES

2

In Sections 2.2 and 2.4, we

investigated infinite limits and

vertical asymptotes. There, we let x approach a number. The result was that the values of y became arbitrarily

large (positive or negative).

LIMITS AND DERIVATIVES

2.6Limits at Infinity:

Horizontal Asymptotes

LIMITS AND DERIVATIVES

In this section, we:

Let x become arbitrarily large (positive or negative) and see what happens to y.

Let’s begin by investigating the behavior

of the function f defined by

as x becomes large.

2

2

1( )

1

xf x

x

HORIZONTAL ASYMPTOTES

The table gives values of this

function correct to six decimal

places.

The graph of f has been

drawn by a computer in the

figure.


As x grows larger and larger,

you can see that the values of

f(x) get closer and closer to 1. It seems that we can make the

values of f(x) as close as we like to 1 by taking x sufficiently large.


This situation is expressed symbolically

by writing

In general, we use the notation

to indicate that the values of f(x) become

closer and closer to L as x becomes larger

and larger.

lim ( )x

f x L


2

2

1lim 1

1x

x

x

Let f be a function defined on some

interval .

Then,

means that the values of f(x) can be

made arbitrarily close to L by taking x

sufficiently large.

( , )a

lim ( )x

f x L

HORIZONTAL ASYMPTOTES 1. Definition

Another notation for is

as

The symbol does not represent a number. Nonetheless, the expression is often read

as:“the limit of f(x), as x approaches infinity, is L”or “the limit of f(x), as x becomes infinite, is L”or “the limit of f(x), as x increases without bound, is L”

lim ( )x

f x L

( )f x L x


lim ( )x

f x L

The meaning of such phrases is given

by Definition 1.

A more precise definition—similar to

the definition of Section 2.4—is

given at the end of this section.


,

Geometric illustrations of Definition 1

are shown in the figures. Notice that there are many ways for the graph of f to

approach the line y = L (which is called a horizontal asymptote) as we look to the far right of each graph.


Referring to the earlier figure, we see that,

for numerically large negative values of x,

the values of f(x) are close to 1. By letting x decrease through negative values without

bound, we can make f(x) as close as we like to 1.


This is expressed by writing

The general definition is as follows.

2

2

1lim 1

1x

x

x


Let f be a function defined on some

interval .

Then,

means that the values of f(x) can be

made arbitrarily close to L by taking x

sufficiently large negative.

( , )a

lim ( )x

f x L


Again, the symbol does not

represent a number.

However, the expression

is often read as:

“the limit of f(x), as x approaches

negative infinity, is L”

lim ( )x

f x L


Definition 2

is illustrated in

the figure. Notice that the graph

approaches the line y = L as we look to the far left of each graph.


The line y = L is called a horizontal

asymptote of the curve y = f(x) if either

lim ( ) or lim ( )x x

f x L f x L


For instance, the curve illustrated in

the earlier figure has the line y = 1 as

a horizontal asymptote because2

2

1lim 1

1x

x

x


An example of a curve with two

horizontal asymptotes is y = tan-1x.


In fact,

So, both the lines and are horizontal asymptotes.

This follows from the fact that the lines are vertical asymptotes of the graph of tan.

2y 2y

2x

1 1lim tan lim tan2 2x x

x x


Find the infinite limits, limits at infinity,

and asymptotes for the function f whose

graph is shown in the figure.

HORIZONTAL ASYMPTOTES Example 1

We see that the values of f(x) become

large as from both sides.

So,

1x

limx 1

f (x)


Notice that f(x) becomes large negative

as x approaches 2 from the left, but large

positive as x approaches 2 from the right. So,

Thus, both the lines x = -1 and x = 2 are vertical asymptotes.

2 2lim ( ) and lim ( )x x

f x f x


As x becomes large, it appears that f(x)

approaches 4.

However, as x decreases through negative

values, f(x) approaches 2. So,

and

This means that both y = 4 and y = 2 are horizontal asymptotes.

lim ( ) 4x

f x


lim ( ) 2x

f x

Find and

Observe that, when x is large, 1/x is small. For instance,

In fact, by taking x large enough, we can make 1/x as close to 0 as we please.

Therefore, according to Definition 1, we have

1limx x

1limx x

1 1 10.01 , 0.0001 , 0.000001

100 10,000 1,000,000


1lim 0x x

Similar reasoning shows that, when x

is large negative, 1/x is small negative.

So, we also have It follows that the line y = 0 (the x-axis) is a horizontal

asymptote of the curve y = 1/x. This is an equilateral hyperbola.

1lim 0x x


Most of the Limit Laws given

in Section 2.3 also hold for limits

at infinity. It can be proved that the Limit Laws (with the exception

of Laws 9 and 10) are also valid if is replaced by or .

In particular, if we combine Laws 6 and 11 with the results of Example 2, we obtain the following important rule for calculating limits.

x ax x


If r > 0 is a rational number, then

If r > 0 is a rational number such that xr

is defined for all x, then

1lim 0

rx x

1lim 0

rx x

HORIZONTAL ASYMPTOTES 5. Theorem

Evaluate

and indicate which properties of limits

are used at each stage.

As x becomes large, both numerator and denominator become large.

So, it isn’t obvious what happens to their ratio. We need to do some preliminary algebra.

2

2

3 2lim

5 4 1x

x x

x x


To evaluate the limit at infinity of any rational

function, we first divide both the numerator

and denominator by the highest power of x

that occurs in the denominator. We may assume that , since we are interested

in only large values of x.0x


In this case, the highest power of x in the

denominator is x2. So, we have:2

2 2 2

2 2

22

3 2 1 23

3 2lim lim lim

4 15 4 1 5 4 1 5x x x

x xx x xx xx x x x

x xx


3 0 0(by Limit Law 7 and Theoreom 5)

5 0 03

5

2

2

1 1lim3 lim 2lim

(by Limit Laws 1, 2, and 3)1 1

lim5 4 lim lim

x x x

x x x

x x

x x


2

2

1 2lim 3

(by Limit Law 5)4 1

lim 5

x

x

x x

x x

A similar calculation shows that the limit

as is also

The figure illustrates the results of these calculations by showing how the graph of the given rational function approaches the horizontal asymptote

x 3

5

3

5y


Find the horizontal and vertical

asymptotes of the graph of the

function22 1

( )3 5

xf x

x


Dividing both numerator and denominator

by x and using the properties of limits,

we have:


2 22

12

2 1lim lim (since for 0)

53 5 3x x

x x x x xx

x

2 2

1 1lim 2 lim 2 lim

2 0 215 3 5.0 3lim3 5limlim 3

x x x

x xx

x x

xx

Therefore, the line is

a horizontal asymptote of the graph of f.

2 / 3y HORIZONTAL ASYMPTOTES Example 4

In computing the limit as ,

we must remember that, for x < 0,

we have

So, when we divide the numerator by x, for x < 0, we get

Therefore,

x

2x x x

2 222

1 1 12 1 2 1 2x x

x xx

limx

2x2 1

3x 5 lim

x

2 1

x2

3 5

x

2 lim

x

1

x2

3 5 limx

1

x

2

3


Thus, the line is also

a horizontal asymptote.

23y


A vertical asymptote is likely to occur

when the denominator, 3x - 5, is 0,

that is, when

If x is close to and , then the denominator is close to 0 and 3x - 5 is positive.

The numerator is always positive, so f(x) is positive.

Therefore,

5

3x 5

35

3x

22 1x


2

(5 3)

2 1lim

3 5x

x

x

If x is close to but , then 3x – 5 < 0, so f(x) is large negative.

Thus,

The vertical asymptote is

5

35

3x

2

(5 3)

2 1lim

3 5x

x

x

5

3x


Compute

As both and x are large when x is large, it’s difficult to see what happens to their difference.

So, we use algebra to rewrite the function.

2lim 1x

x x

2 1x


We first multiply the numerator and

denominator by the conjugate radical:

The Squeeze Theorem could be used to show that this limit is 0.

2

2 2

2

2 2

2 2

1lim 1 lim 1

1

( 1) 1lim lim

1 1

x x

x x

x xx x x x

x x

x x

x x x x


However, an easier method is to divide the numerator and denominator by x.

Doing this and using the Limit Laws, we obtain:

2

2 2

2

11

lim 1 lim lim1 1

10

lim 01 1 0 1

1 1

x x x

x

xx xx x x x

x

x

x


The figure illustrates this

result.


The graph of the natural exponential

function y = ex has the line y = 0

(the x-axis) as a horizontal asymptote. The same is true of any exponential function with

base a > 1.


In fact, from the graph in the figure

and the corresponding table of values,

we see that Notice that the values of ex approach 0 very rapidly.

lim 0x

xe


Evaluate

If we let t = 1/x, we know that as

Therefore, by (6),

1

0lim x

xe

0x t

1

0lim lim 0tx

txe e


Evaluate

As x increases, the values of sin x oscillate between 1 and -1 infinitely often.

So, they don’t approach any definite number. Thus, does not exist.

limsinx

x


limsinx

x

The notation is used to

indicate that the values of f(x) become

large as x becomes large. Similar meanings are attached to the following symbols:

lim ( )x

f x

lim ( )x

f x

INFINITE LIMITS AT INFINITY

lim ( )x

f x

lim ( )x

f x

Find and

When x becomes large, x3 also becomes large.

For instance,

In fact, we can make x3 as big as we like by taking x large enough.

Therefore, we can write

3limx

x

3limx

x

3 3 310 1,000 100 1,000,000 1,000 1,000,000,000

3limx

x

Example 8INFINITE LIMITS AT INFINITY

Similarly, when x is large negative, so is x3. Thus,

These limit statements can also be seen from the graph of y = x3 in the figure.

3limx

x


Looking at the first figure, we see

However, as the second figure demonstrates,

y = ex becomes large as at a much

faster rate than y = x3.

lim x

xe

x


Find

It would be wrong to write

The Limit Laws can’t be applied to infinite limits because is not a number ( can’t be defined).

However, we can write

This is because both x and x - 1 become arbitrarily large and so their product does too.

2lim( )x

x x

2 2lim( ) lim limx x x

x x x x

2lim( ) lim ( 1)x x

x x x x


Find

As in Example 3, we divide the numerator and denominator by the highest power of x in the denominator, which is just x:

because and as

2

lim3x

x x

x

2 1lim lim

33 1x x

x x x

xx

1x 3 1 1x x


The next example shows that, by using

infinite limits at infinity, together with

intercepts, we can get a rough idea of the

graph of a polynomial without having to plot

a large number of points.


Sketch the graph of

by finding its intercepts and its limits

as and as

The y-intercept is f(0) = (-2)4(1)3(-1) = -16 The x-intercepts are found by setting y = 0: x = 2, -1, 1.

4 3( 2) ( 1) ( 1)y x x x

x x


Notice that, since (x - 2)4 is positive,

the function doesn’t change sign at 2.

Thus, the graph doesn’t cross the x-axis

at 2. It crosses the axis at -1 and 1.


When x is large positive, all three factors

are large, so

When x is large negative, the first factor

is large positive and the second and third

factors are both large negative, so4 3lim ( 2) ( 1) ( 1)

xx x x

4 3lim( 2) ( 1) ( 1)x

x x x


Combining this information,

we give a rough sketch of the graph

in the figure.


Definition 1 can be stated precisely as follows.

Let f be a function defined on some interval (a, ). Then, means that, for every ,

there is a corresponding number N such thatif x > N, then

lim ( )x

f x L

0

( )f x L

PRECISE DEFINITIONS 7. Definition

In words, this says that the values of f(x) can

be made arbitrarily close to L (within a

distance , where is any positive number)

by taking x sufficiently large (larger than N,

where N depends on ).

PRECISE DEFINITIONS

Graphically, it says that, by choosing x large

enough (larger than some number N), we can

make the graph of f lie between the given

horizontal lines and This must be true no matter how small we choose .

y L y L

PRECISE DEFINITIONS

This figure shows that, if a smaller value

of is chosen, then a larger value of N

may be required.

PRECISE DEFINITIONS

Similarly, a precise version of Definition 2

is given as follows.

Let f be a function defined on some interval

( ,a).

Then, means that, for every ,

there is a corresponding number N such that,

if x < N, then

lim ( )x

f x L

0

( )f x L


This is illustrated in the

figure.

PRECISE DEFINITIONS

In Example 3, we calculated that

In the next example, we use

a graphing device to relate this statement

to Definition 7 with and

2

2

3 2 3lim

5 4 1 5x

x x

x x

3

5L 0.1

PRECISE DEFINITIONS

Use a graph to find a number N

such that, if x > N, then

We rewrite the given inequality as:

PRECISE DEFINITIONS Example 12

2

2

3 20.6 0.1

5 4 1

x x

x x

2

2

3 20.5 0.7

5 4 1

x x

x x

We need to determine the values of x

for which the given curve lies between

the horizontal lines y = 0.5 and y = 0.7 So, we graph the curve and

these lines in the figure.


Then, we use the cursor to estimate

that the curve crosses the line y = 0.5

when To the right of this number, the curve stays between

the lines y = 0.5 and y = 0.7

6.7x


Rounding to be safe, we can say that,

if x > 7, then

In other words, for , we can choose N = 7 (or any larger number) in Definition 7.

2

2

3 20.6 0.1

5 4 1

x x

x x

0.1


Use Definition 7 to prove

Given , we want to find N such that, if x > N, then

In computing the limit, we may assume that x > 0 Then,

1lim 0x x

0 1

0x

1 1xx


Let’s choose

So, if , then

Therefore, by Definition 7,

1N

1x N

1 10

x x


1lim 0x x

The figure illustrates the proof by

showing some values of and the

corresponding values of N.


Finally, we note that an infinite limit at infinity

can be defined as follows.

Let f be a function defined on some interval

(a, ).

Then, means that, for every

positive number M, there is a corresponding

positive number N such that,

if x > N, then f(x) > M

lim ( )x

f x


The geometric illustration is

given in the figure. Similar definitions apply when the symbol

is replaced by

PRECISE DEFINITIONS

LIMITS AND DERIVATIVES 2. In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The.

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