APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree....
-
Upload
norah-carpenter -
Category
Documents
-
view
214 -
download
2
Transcript of APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree....
![Page 1: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/1.jpg)
APPLICATIONS OF DIFFERENTIATIONAPPLICATIONS OF DIFFERENTIATION
4
![Page 2: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/2.jpg)
A polynomial behaves near infinity as its term of highest degree. The polynomial
3425 34 xxx
behaves like the polynomial 45x
Near infinity
![Page 3: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/3.jpg)
4.4Limits at Infinity;
Horizontal Asymptotes
In this section, we will learn about:
Various aspects of horizontal asymptotes.
APPLICATIONS OF DIFFERENTIATION
![Page 4: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/4.jpg)
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
![Page 5: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/5.jpg)
2
2
1( )
1
xf x
x
HORIZONTAL ASYMPTOTES
11
12
2
2
2
x
x
x
x
So f(x) behaves like 1 when x is near infinity
Near infinity, one can write,
![Page 6: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/6.jpg)
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
![Page 7: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/7.jpg)
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
![Page 8: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/8.jpg)
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
![Page 9: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/9.jpg)
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
![Page 10: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/10.jpg)
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
![Page 11: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/11.jpg)
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
![Page 12: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/12.jpg)
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
![Page 13: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/13.jpg)
This is expressed by writing
The general definition is as follows.
2
2
1lim 1
1x
x
x
HORIZONTAL ASYMPTOTES
![Page 14: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/14.jpg)
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
![Page 15: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/15.jpg)
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
![Page 16: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/16.jpg)
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
![Page 17: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/17.jpg)
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
![Page 18: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/18.jpg)
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
![Page 19: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/19.jpg)
The curve y = f(x) sketched here has both
y = -1 and y = 2 as horizontal asymptotes.
This is because:
HORIZONTAL ASYMPTOTES
lim 1 and lim 2x x
f x f x
![Page 20: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/20.jpg)
Find the infinite limits, limits at infinity,
and asymptotes for the function f whose
graph is shown in the figure.
HORIZONTAL ASYMPTOTES Example 1
![Page 21: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/21.jpg)
We see that the values of f(x) become
large as from both sides.
So,
1x
limx 1
f (x)
HORIZONTAL ASYMPTOTES Example 1
![Page 22: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/22.jpg)
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
![Page 23: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/23.jpg)
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
![Page 24: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/24.jpg)
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
![Page 25: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/25.jpg)
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
![Page 26: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/26.jpg)
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
![Page 27: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/27.jpg)
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
![Page 28: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/28.jpg)
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
![Page 29: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/29.jpg)
Quick Method
2
2
3 2lim
5 4 1x
x x
x x
HORIZONTAL ASYMPTOTES Example 3
5
3
5
3
145
232
2
2
2
x
x
xx
xx
Near Infinity, one can write
![Page 30: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/30.jpg)
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
![Page 31: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/31.jpg)
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
![Page 32: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/32.jpg)
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
![Page 33: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/33.jpg)
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
![Page 34: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/34.jpg)
Find the horizontal and vertical
asymptotes of the graph of the
function22 1
( )3 5
xf x
x
HORIZONTAL ASYMPTOTES Example 4
![Page 35: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/35.jpg)
Quick Method22 1
( )3 5
xf x
x
HORIZONTAL ASYMPTOTES Example 4
x
x
x
x
x
x
x
x ||
3
2
3
2
3
2
53
12 222
3
2
3
||2
x
x
3
2
3
||2
x
xIf x>0 and If x <0
For x near infinity, one can write
![Page 36: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/36.jpg)
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
![Page 37: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/37.jpg)
Therefore, the line is
a horizontal asymptote of the graph of f.
2 / 3y HORIZONTAL ASYMPTOTES Example 4
![Page 38: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/38.jpg)
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
![Page 39: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/39.jpg)
Thus, the line is also
a horizontal asymptote.
23y
HORIZONTAL ASYMPTOTES Example 4
![Page 40: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/40.jpg)
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
![Page 41: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/41.jpg)
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
![Page 42: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/42.jpg)
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
![Page 43: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/43.jpg)
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
![Page 44: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/44.jpg)
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
For x near infinity, one can write that
02
1
||
11
1
122
xxxxxxxx
![Page 45: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/45.jpg)
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
![Page 46: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/46.jpg)
The figure illustrates this
result.
HORIZONTAL ASYMPTOTES Example 5
![Page 47: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/47.jpg)
Evaluate
If we let t = 1/x, then as .
Therefore, .
1lim sin x x
0t x
0
1lim sin lim sin 0
x tt
x
HORIZONTAL ASYMPTOTES Example 6
![Page 48: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/48.jpg)
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
![Page 49: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/49.jpg)
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
![Page 50: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/50.jpg)
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
![Page 51: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/51.jpg)
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
![Page 52: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/52.jpg)
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
![Page 53: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/53.jpg)
Find 2lim( )
xx x
Example 9INFINITE LIMITS AT INFINITY
xxxx 22
Near infinity, one can write:
![Page 54: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/54.jpg)
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
![Page 55: APPLICATIONS OF DIFFERENTIATION 4. A polynomial behaves near infinity as its term of highest degree. The polynomial behaves like the polynomial Near infinity.](https://reader034.fdocuments.in/reader034/viewer/2022050714/56649eaa5503460f94baf7ba/html5/thumbnails/55.jpg)
Find
2
lim3x
x x
x
Example 10INFINITE LIMITS AT INFINITY
xx
x
x
x
xx 22
3
Near infinity, one can write