limits and derivatives 2. in sections 2.2 and 2.4, we investigated infinite limits and vertical...
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LIMITS AND DERIVATIVESLIMITS AND DERIVATIVES
2
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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
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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.
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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
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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.
HORIZONTAL ASYMPTOTES
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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.
HORIZONTAL ASYMPTOTES
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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
HORIZONTAL ASYMPTOTES
2
2
1lim 1
1x
x
x
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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
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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
HORIZONTAL ASYMPTOTES
lim ( )x
f x L
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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.
HORIZONTAL ASYMPTOTES
,
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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.
HORIZONTAL ASYMPTOTES
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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.
HORIZONTAL ASYMPTOTES
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This is expressed by writing
The general definition is as follows.
2
2
1lim 1
1x
x
x
HORIZONTAL ASYMPTOTES
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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
HORIZONTAL ASYMPTOTES 2. Definition
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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
HORIZONTAL ASYMPTOTES
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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.
HORIZONTAL ASYMPTOTES
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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
HORIZONTAL ASYMPTOTES 3. Definition
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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
HORIZONTAL ASYMPTOTES 3. Definition
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An example of a curve with two
horizontal asymptotes is y = tan-1x.
HORIZONTAL ASYMPTOTES
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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
HORIZONTAL ASYMPTOTES 4. Definition
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Find the infinite limits, limits at infinity,
and asymptotes for the function f whose
graph is shown in the figure.
HORIZONTAL ASYMPTOTES Example 1
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We see that the values of f(x) become
large as from both sides.
So,
1x
limx 1
f (x)
HORIZONTAL ASYMPTOTES Example 1
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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
HORIZONTAL ASYMPTOTES Example 1
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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
HORIZONTAL ASYMPTOTES Example 1
lim ( ) 2x
f x
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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
HORIZONTAL ASYMPTOTES Example 2
1lim 0x x
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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
HORIZONTAL ASYMPTOTES Example 2
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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
HORIZONTAL ASYMPTOTES
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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
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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
HORIZONTAL ASYMPTOTES Example 3
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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
HORIZONTAL ASYMPTOTES Example 3
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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
HORIZONTAL ASYMPTOTES Example 3
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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
HORIZONTAL ASYMPTOTES Example 3
2
2
1 2lim 3
(by Limit Law 5)4 1
lim 5
x
x
x x
x x
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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
HORIZONTAL ASYMPTOTES Example 3
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Find the horizontal and vertical
asymptotes of the graph of the
function22 1
( )3 5
xf x
x
HORIZONTAL ASYMPTOTES Example 4
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Dividing both numerator and denominator
by x and using the properties of limits,
we have:
HORIZONTAL ASYMPTOTES Example 4
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
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Therefore, the line is
a horizontal asymptote of the graph of f.
2 / 3y HORIZONTAL ASYMPTOTES Example 4
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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
HORIZONTAL ASYMPTOTES Example 4
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Thus, the line is also
a horizontal asymptote.
23y
HORIZONTAL ASYMPTOTES Example 4
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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
HORIZONTAL ASYMPTOTES Example 4
2
(5 3)
2 1lim
3 5x
x
x
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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
HORIZONTAL ASYMPTOTES Example 4
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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
HORIZONTAL ASYMPTOTES Example 5
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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
HORIZONTAL ASYMPTOTES Example 5
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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
HORIZONTAL ASYMPTOTES Example 5
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The figure illustrates this
result.
HORIZONTAL ASYMPTOTES Example 5
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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.
HORIZONTAL ASYMPTOTES
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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
HORIZONTAL ASYMPTOTES 6. Definition
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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
HORIZONTAL ASYMPTOTES Example 6
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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
HORIZONTAL ASYMPTOTES Example 7
limsinx
x
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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
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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
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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
Example 8INFINITE LIMITS AT INFINITY
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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
INFINITE LIMITS AT INFINITY
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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
Example 9INFINITE LIMITS AT INFINITY
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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
Example 10INFINITE LIMITS AT INFINITY
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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.
INFINITE LIMITS AT INFINITY
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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
Example 11INFINITE LIMITS AT INFINITY
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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.
Example 11INFINITE LIMITS AT INFINITY
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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
Example 11INFINITE LIMITS AT INFINITY
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Combining this information,
we give a rough sketch of the graph
in the figure.
Example 11INFINITE LIMITS AT INFINITY
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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
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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
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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
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This figure shows that, if a smaller value
of is chosen, then a larger value of N
may be required.
PRECISE DEFINITIONS
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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
PRECISE DEFINITIONS 8. Definition
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This is illustrated in the
figure.
PRECISE DEFINITIONS
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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
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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
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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.
PRECISE DEFINITIONS Example 12
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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
PRECISE DEFINITIONS Example 12
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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
PRECISE DEFINITIONS Example 12
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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
PRECISE DEFINITIONS Example 13
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Let’s choose
So, if , then
Therefore, by Definition 7,
1N
1x N
1 10
x x
PRECISE DEFINITIONS Example 13
1lim 0x x
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The figure illustrates the proof by
showing some values of and the
corresponding values of N.
PRECISE DEFINITIONS Example 13
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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
PRECISE DEFINITIONS 9. Definition
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The geometric illustration is
given in the figure. Similar definitions apply when the symbol
is replaced by
PRECISE DEFINITIONS