Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:
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Transcript of Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:
![Page 1: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/1.jpg)
Algebra of LimitsAssume that both of the following limits exist and c and is a real number:
Then:
0)(lim
),(lim/)(lim)(/)(lim..4
)(lim)(lim)()(lim..3
)(lim)(lim)()(lim..2
)(lim)(lim..1
xgthatprovided
xgxfxgxf
xgxfxgxf
xgxfxgxf
xfcxcf
ax
axaxax
axaxax
axaxax
axax
![Page 2: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/2.jpg)
Calculating LimitsFinding the limit of a function f a point x = a.
Distinguishing the following cases:1. The case when f is continuous a x = a.2. The case 0/0.3. The case ∞/ ∞4. The case of an infinite limit5. The case c/∞, where c is a real number.6. The case, when it is possible to use the
squeeze theorem.
![Page 3: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/3.jpg)
1. The case when f is continuous at x = a
If f is continues at x=a, then:
Notice:1. Polynomial functions and the cubic root function ( & all functions of its two families) are everywhere continuous.2. Rational, trigonometric and root functions are continuous at every point of their domains.3. If f and g are continuous a x=a, then so are cf, f+g, f-g, fg and f/g (provided that he limit of f at x=a is not zero)
![Page 4: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/4.jpg)
Examples for the case when f is continuous at x = a
04
0
22
4)2()2()(lim
2.},2{
.2
4)(
2
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)1(
2
2
2
2
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fxf
xatcontisitsoandRon
contisx
xxffunctionrationalThe
x
x
Example
x
x
![Page 5: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/5.jpg)
Examples for the case when f is continuous at x = a
22)2(248)321(
3191)3)1(2)1(()1()(lim
,
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9
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fxf
soand
xatcontisxxxxxffunctionThe
Thus
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Example
x
x
![Page 6: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/6.jpg)
Examples for the case when f is continuous at x= a
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![Page 7: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/7.jpg)
2. The case 0/0
Suppose we want to find:
For the case when:
Then this is called the case 0/0. Caution: The limit is not equal 0/0. This is just a name that classifies the type of limits having such property.
)(
)(lim
xh
xgax
.0)(lim&)(lim arexhxgaxax
![Page 8: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/8.jpg)
Examples for the case 0/0
8
3
32
12
)8(4
444
)4)(2(
42lim
)4)(2)(2(
)42)(2(lim
16
8lim
16lim08lim
:,0/016
8lim
:)1(
2
2
2
2
2
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4
3
2
4
2
3
2
4
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2
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becausecasetheisThisx
x
factoringbySolvingExample
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x
x
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x
![Page 9: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/9.jpg)
Examples for the case 0/0
30
1
)55(3
1
525)3(
1lim
525)3(lim
525)3(
25)25(lim
525
525.
)3(
525lim
)3(
525lim
)3(lim0525lim
:,0/0
)3(
525lim
)2(
0
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becausecasetheisThis
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methodconjugatethebygMultiplyinExample
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![Page 10: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/10.jpg)
Examples for the case 0/0
existnotDoesx
xx
x
x
xThus
x
x
x
x
x
x
x
x
x
x
x
x
x
thatNotice
xx
becausecasetheisThis
Solutionx
xexistsifFind
ValuesAbsoluteInvolvingExample
x
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x
153
102lim
153
102lim
153
102lim:
3
2
3
2lim
)5(3
)5(2lim
153
)102(lim
153
102lim
3
2
3
2lim
)5(3
)5(2lim
153
102lim
153
102lim
102
153lim0102lim
:,0/0
:153
102lim:
)3(
5
)5()5(
)5()5()5()5(
)5()5()5()5(
5;102
5;)102(
55
5
![Page 11: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/11.jpg)
Question: Simplify the formula of f and graph it!
5;3
2
5;3
2
5;153
102
5;153
)102(153
102)(
x
x
xx
x
xx
xx
xxf
![Page 12: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/12.jpg)
Questions
fGraphc
fofformulatheSimplifyb
xpotheatittheand
itlefttheitlefttheexistseitherifFinda
x
xfLet
fGraphc
fofformulatheSimplifyb
xpotheatittheand
itlefttheitlefttheexistseitherifFinda
x
xfLet
.
.
.0intlim
lim,lim,,.
3
62.2
.
.
.0intlim
lim,lim,,.
.1
![Page 13: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/13.jpg)
3. The case ∞/ ∞
Suppose we want to find:
For the case when the limits of both functions f and g are infinite
Then this is called the case ∞/ ∞. Caution: The limit is not equal ∞/∞. This is just a name that classifies the type of limits having such property.
)(
)(lim
)(
)(lim
xh
xgOr
xh
xgxax
![Page 14: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/14.jpg)
Limits at infinity
A function y=f(x) may approach a real number b as x increases or decreases with no bound.When this happens, we say that f has a limit at infinity, and that the line y=b is a horizontal asymptote for f.
![Page 15: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/15.jpg)
Limit at infinity: The Case of Rational Functions
A rational function r(x) = p(x)/q(x) has a limit at infinity if the degree of p(x) is equal or less than the degree of q(x).
A rational function r(x) = p(x)/q(x) does not have a limit at infinity (but has rather infinite right and left limits) if the degree of p(x) is greater than the degree of q(x).
![Page 16: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/16.jpg)
Example (1)Let
Find
Solution:Since the degree of the polynomial in the numerator, which is 9, is equal to the degree of the polynomial in the denominator, then
146
325)(
79
29
xx
xxxf
)(lim xfx
![Page 17: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/17.jpg)
To show that, we follow the following steps:
6
5
)(lim9
9
inatordenomtheinxofcofficientThe
numeratortheinxofcofficientThexf
x
6
5
0)0(46
)0(3)0(25
1lim
1lim46lim
1lim3
1lim25lim
1146lim
3125lim
1146
3125
lim146
325lim)(lim
92
97
92
97
92
97
79
29
xx
xx
xx
xx
xx
xxxx
xxxf
xxx
xxx
x
x
xxx
![Page 18: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/18.jpg)
Example (2)Let
Find
Solution:Since the degree of the polynomial in the numerator, which is 9, is less than the degree of the polynomial in the denominator, which is 12, then
146
325)(
712
29
xx
xxxf
)(lim xfx
![Page 19: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/19.jpg)
To show that, we follow the following steps:
0)(lim
xfx
06
0
0)0(46
)0(3)0(2)0(5
1lim
1lim46lim
1lim3
1lim2
1lim5
1146lim
312
5lim
1146
312
5
lim146
325lim)(lim
125
12103
125
12103
125
12103
712
29
xx
xxx
xx
xxx
xx
xxxxx
xxxf
xxx
xxx
x
x
xxx
![Page 20: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/20.jpg)
Example (3)Let
Find
Solution:Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 9, then
146
325)(
79
212
xx
xxxf
)(lim xfx
![Page 21: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/21.jpg)
They are infinite limits. To show that, we follow the following steps:
3
9
12
79
212
lim
6
5lim
146
325lim)(lim
xassamethearewhich
x
xassametheare
xx
xxxf
x
x
xx
.)(lim existnotdoxfx
![Page 22: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/22.jpg)
Example (4)Let
Find
Solution:Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 9, then
146
325)(
79
212
xx
xxxf
)(lim xfx
![Page 23: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/23.jpg)
They are infinite limits. To show that, we follow the following steps:
)(lim
6
5lim
146
325lim)(lim
3
9
12
79
212
xassamethearewhich
x
xassametheare
xx
xxxf
x
x
xx
.)(lim existnotdoxfx
![Page 24: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/24.jpg)
Example (5)Let
Find
Solution:Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 8, then
146
325)(
78
212
xx
xxxf
)(lim xfx
![Page 25: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/25.jpg)
They are infinite limits. To show that, we follow the following steps:
.)(lim existnotdoxfx
4
8
12
78
212
lim
6
5lim
146
325lim)(lim
xassamethearewhich
x
xassametheare
xx
xxxf
x
x
xx
![Page 26: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/26.jpg)
Example (6)Let
Find
Solution:Since the degree of the polynomial in the numerator, which is 12, is greater than the degree of the polynomial in the denominator, which is 8, then
146
325)(
78
212
xx
xxxf
)(lim xfx
![Page 27: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/27.jpg)
They are infinite limits. To show that, we follow the following steps:
.)(lim existnotdoxfx
)(lim
6
5lim
146
325lim)(lim
4
8
12
78
212
xassamethearewhich
x
xassametheare
xx
xxxf
x
x
xx
![Page 28: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/28.jpg)
Limits & Infinity
Problems Involving Roots
![Page 29: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/29.jpg)
Introduction
We know that:
√x2 = |x|, which is equal x is x non-negative and equal to – x if x is negative
For if x = 2, then √(2)2 = √4 = 2 = |2|& if x = - 2, then √(-2)2 = √4 = -(-2) =|-2|
![Page 30: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/30.jpg)
Example
0;16
0;3
0;16
00;)16(
0;3
00;16
0;3
0;)16(
0;3
2
2
29
29
29
292
292
916)(
:
916)(
:
xx
x
xx
xorxx
x
xorxx
x
xx
x
x
x
x
xx
xxf
asrewwrittenbecan
xxf
Let
![Page 31: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/31.jpg)
Example
fofasymptoteshorizontaltheareyandylinestheThus
Whyx
x
x
x
xx
xx
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x
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x
xx
x
x
xxf
formulaoneintogetheritsmlithesefindwillWe
atandatfofitsmlithefindfrstWe
Solution
fofasymptoteshorizontaltheFindx
xxf
Let
xx
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22
?2
2
4
02
0162
lim2lim
9lim16lim
22
916
lim)
22(
916
lim22
916
lim
22
916
lim22
)9
16(lim
22
916lim)(lim
,
:
22
916)(
:
2
222
22
22
2
2
![Page 32: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/32.jpg)
4. The case of infinite limit
![Page 33: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/33.jpg)
Infinite Limits
A function f may increases or decreases with no bound near certain values c for the independent variable x. When this happens, we say that f has an infinite limit, and that f has a vertical asymptote at x = c The line x=c is called a vertical asymptote for f.
![Page 34: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/34.jpg)
Infinite Limits
A function has an infinite one-sided limit at a point x=c if at that point the considered one-sided limit of the denominator is zero and that of the numerator is not zero. The sign of the infinite limit is determined by the sign of both the numerator and the denominator at values close to the considered point x=c approached by the variable x (from the considered side).
![Page 35: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/35.jpg)
Infinite Limits- The Case of Rational Functions
A rational function has an infinite one-sided limit at a point x=c if c a zero of the denominator but not of the numerator. The sign of the infinite limit is determined by the sign of both the numerator and the denominator at values close to the considered point x=c approached by the variable x (from the considered side).
![Page 36: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/36.jpg)
Example (1)Let
Find
Solution:First x=0 is a zero of the denominator which is not a zero of the numerator.
xxf
1)(
)(lim.0
xfax
)(lim.0
xfbx
![Page 37: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/37.jpg)
a. As x approaches 0 from the right, the numerator is always positive ( it is equal to 1) and the denominator approaches 0 while keeping positive; hence, the function increases with no bound. Thus:
)(lim0
xfx
The function has a vertical asymptote at x = 0, which is the line x = 0 (see the graph in the file on basic algebraic functions).
b. As x approaches 0 from the left, the numerator is always positive ( it is equal to 1) and the denominator approaches 0 while keeping negative; hence, the function decreases with no bound.
)(lim0
xfx
![Page 38: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/38.jpg)
Example (2)Let
Find
Solution:First x=1 is a zero of the denominator which is not a zero of the numerator.
)(lim.1
xfax
)(lim.1
xfbx
1
5)(
x
xxf
![Page 39: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/39.jpg)
)(lim1
xfx
The function has a vertical asymptote at x = 1, which is the line x = 1
)(lim1
xfx
a. As x approaches 1 from the right, the numerator approaches 6 (thus keeping positive), and the denominator approaches 0 while keeping positive; hence, the function increases with no bound. Thus,
b. As x approaches 1 from the left, the numerator approaches 6 (thus keeping positive), and the denominator approaches 0 while keeping negative; hence, the function decreases with no bound. The function has a vertical asymptote at x=1, which is the line x = 1. Thus:
![Page 40: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/40.jpg)
Example (3)Let
Find
Solution:First x=3 is a zero of the denominator which is not a zero of the numerator.
)(lim.3
xfax
)(lim.3
xfbx
)3)(1(
)4)(1(
34
45)(
2
2
xx
xx
xx
xxxf
![Page 41: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/41.jpg)
)(lim
)033,3(
,03
;14343
3
4lim
)3)(1(
)4)(1(lim
34
45lim
)(lim
3
3
3
2
2
3
3
xfThus
xsoandxhavewexasbecause
positivekeepingwhilexand
negativekeepingthusxxAs
x
x
xx
xxxx
xx
xf
x
x
x
x
x
![Page 42: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/42.jpg)
)(lim
)033,3(
,03
;14343
3
4lim
)3)(1(
)4)(1(lim
34
45lim
)(lim
3
3
3
2
2
3
3
xfThus
xsoandxhavewexasbecause
negativekeepingwhilexand
negativekeepingthusxxAs
x
x
xx
xxxx
xx
xf
x
x
x
x
x
![Page 43: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/43.jpg)
6. The case constant/∞Suppose we want to find:
For the case when:
In this case, no mater what the formulas of g and h are, we will always have:
Then this is called the case c/∞. Caution: The limit is not equal c/ ∞. This is just a name that classifies the type of limits having such property. This limit is always equal zero
)(
)(lim
xh
xgax
)(lim&)(lim xhRcxgaxax
0)(
)(lim
xh
xgax
![Page 44: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/44.jpg)
Example on the case constant/∞
01
1lim
)(
)(lim
,
?)()(lim1)(
11lim)(lim1)(
:
:
1
1lim
,
:)1(
xxxh
xg
Thus
Whyxhxxxh
xgxg
haveWe
Solution
xx
Find
Example
xx
x
xx
x
![Page 45: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/45.jpg)
6. Using the Squeeze Theorem
![Page 46: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/46.jpg)
The Squeeze (Sandwich or Pinching)) Theorem
Suppose that we want to find the limit of a function f at a given point x=a and that the values of f on some interval containing this point (with the possible exception of that point) lie between the values of a couple of functions g and h whose limits at x=a are equal. The squeeze theorem says that in this case the limit of f at x=a will equal the limit of g and h at this point.
![Page 47: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/47.jpg)
The Squeeze Theory
lxf
Then
xhlxg
dcawhere
adcxxhxfxg
Let
ax
axax
)(
:
)()(
&
),(
),(;)()()(
:
lim
limlim
![Page 48: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/48.jpg)
Example (1)
1)(
,,
)4,0(;)(12,)4,0(1
&
1)1(&112)12(
:
:
)(
)4,0(;)(12
:
lim
limlim
lim
1
2
22
11
1
2
xf
theoremsqueezethebyThus
xxxfx
xx
haveWe
Soluion
xf
Find
xxxfx
Let
x
xx
x
![Page 49: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/49.jpg)
Example (2)
7)(
,,
),0(;)(12,),0(4
&
771616)74(
,7916)94(
:
:
)(
),0[;74)(94
:
lim
lim
lim
lim
4
2
2
4
4
4
2
xf
theoremsqueezethebyThus
xxxfx
xx
x
haveWe
Soluion
xf
Find
xxxxfx
Let
x
x
x
x
![Page 50: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/50.jpg)
Example (3)
0)1
sin(
,
)(0)(
'
0)5,5(;1
sin
)...(
?0)5,5(;11
sin1
:
:
:
)1
sin(
2
0
2
0
2
0
222
22
2
0
lim
limlim
lim
xx
theoremsqueeztheBy
xx
haveWe
xxx
xx
negativenonisxthatNotexbygMultiplyin
Whyxx
haveWe
Soluion
xx
Find
x
xx
x
![Page 51: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/51.jpg)
Example (4)
0)2
cos(
,
)(0)(
'
0)5,5(;2
cos
)...(
?0)5,5(;12
cos1
:
:
:
)2
cos(
4
0
4
0
4
0
444
44
4
0
lim
limlim
lim
xx
theoremsqueezetheBy
xx
haveWe
xxx
xx
negativenonisxthatNotexbygMultiplyin
Whyxx
haveWe
Soluion
xx
Find
x
xx
x
![Page 52: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/52.jpg)
Example (5)
07
cos7
,
707
'
}7{),(;77
cos77
}7{),(,7(
}7{),(;17
cos1
:
}7{),(;17
cos1
:
:
:
7cos7
3
7
77
3
3
3
7
lim
limlim
lim
xx
theoremsqueezetheBy
xx
haveWe
xxx
xx
onpositiveiswhichxbygMultiplyin
xx
soand
xx
haveWe
Soluion
xx
Find
x
xx
x
![Page 53: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/53.jpg)
Example (6)
1)(
,
11112)112(
2)3,0(;)(112
)3,0(2
:
)(
2)3,0(;)(112
:
lim
limlim
lim
2
02
22
2
2
2
xf
theoremsqueezetheBy
eexx
and
xexfxand
Soluion
xf
Find
xexfx
Let
x
x
xx
x
x
x
![Page 54: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/54.jpg)
2)( xexh112)( xxg
![Page 55: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/55.jpg)
Example (7)
1)(
,
110coscos
0)2
,2
(;1)(cos
)2
,2
(0
:
)(
0)2
,2
(;1)(cos
:
lim
limlim
lim
0
00
0
xf
theoremsqueezetheBy
x
and
xxfxand
Soluion
xf
Find
xxfx
Let
x
xx
x
![Page 56: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/56.jpg)
Example (8)
1sin
:),8(,
0)2
,2
(;1sin
cos
:
:
sin
lim
lim
0
0
x
x
atarriveweExampleinascontinuingThus
xx
xx
thatshownbecanIt
Soluion
x
x
Find
x
x
![Page 57: Algebra of Limits Assume that both of the following limits exist and c and is a real number: Then:](https://reader030.fdocuments.in/reader030/viewer/2022032606/56649eb75503460f94bc09eb/html5/thumbnails/57.jpg)
Example (9)
1sin
5
)5sin(
:
]0)5([
0,5,
5:
:
5
)5sin(
limlim
limlim
lim
05
55
5
t
t
x
x
Thus
xtSince
approachestapproachesxasThen
xtLet
Soluion
x
x
Find
tx
xx
x