algebra of limits
DESCRIPTION
Algebra of Limits. Assume that both of the following limits exist and c and is a real number: Then:. Calculating Limits Finding the limit of a function f a point x = a. Distinguishing the following cases: T he case when f is continuous a x = a. The case 0/0. The case ∞ / ∞ - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/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](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/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 of the function f defined by a formula
involving absolute values.6. The case c/∞, where c is a real number.7. Other cases: the case ∞- ∞8. The case, when it is possible to use the squeeze
theorem.
![Page 3: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/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](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/4.jpg)
Examples for the case when f is continuous at x = a
040
224)2()2()(lim
2.},2{
.24)(
24lim
)1(
2
2
2
2
2
fxf
xatcontisitsoandRon
contisxxxffunctionrationalThe
xx
Example
x
x
![Page 5: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/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
,.139)32()(
,1.),,3(3)(
1.9)32()(
1.,9)(
1.,32)(
39)32(lim
)2(
9
95
1
95
95
9
5
95
1
fxf
soandxatcontisxxxxxffunctionThe
ThusxatcontisitthusoncontisxxhfunctionrootThe
xatcontisxxxxgfunctionproducttheTherefore
xatcontisitthuseverywherecontisxxsfunctionThe
xatcontisitthuseverywherecontisxxxpfunctionpolynomialThe
xxxx
Example
x
x
![Page 6: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/6.jpg)
Examples for the case when f is continuous at x= a
155515)5(53lim
5.53)(:,5.
).25..1
:(
.5)(3)(
53lim:)3(
5
5
fxxThus
xatcontisxxxfthusxatcontaretheysoand
hGraphxatcontishthatShow
Questions
everywherecontarexxhandxxg
xxExample
x
x
![Page 7: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/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.
)()(limxhxg
ax
.0)(lim&)(lim arexhxgaxax
![Page 8: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/8.jpg)
Examples for the case 0/0
83
3212
)8(4444
)4)(2(42lim
)4)(2)(2()42)(2(lim
168lim
16lim08lim
:,0/0168lim
:)1(
2
2
2
2
2
2
4
3
2
4
2
3
2
4
3
2
xxxx
xxxxxx
xx
xx
becausecasetheisThisxx
factoringbySolvingExample
x
x
x
xx
x
![Page 9: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/9.jpg)
Examples for the case 0/0
301
)55(31
525)3(1lim
525)3(lim
525)3(25)25(lim
525525.
)3(525lim
)3(525lim
)3(lim0525lim
:,0/0)3(525lim
)2(
0
00
00
00
0
xx
xxxx
xxxx
xx
xxx
xxx
xxx
becausecasetheisThisxx
x
methodconjugatethebygMultiplyinExample
x
xx
xx
xx
x
![Page 10: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/10.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
xhxgOr
xhxg
xax
![Page 11: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/11.jpg)
Examples for the case ∞/∞
1. See the examples involving rational functions in the file on the limit a infinity.
2. Examples involving roots: See the following slides
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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 13: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/13.jpg)
1. 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 14: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/14.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
146325)( 79
29
xxxxxf
)(lim xfx
![Page 15: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/15.jpg)
To show that, we follow the following steps:
65
)(lim 9
9
inatordenomtheinxofcofficientThe
numeratortheinxofcofficientThexfx
65
0)0(46)0(3)0(25
1lim1lim46lim
1lim31lim25lim
1146lim
3125lim
1146
3125lim
146325lim)(lim
92
97
92
97
92
97
79
29
xx
xx
xx
xx
xx
xxxxxxxf
xxx
xxx
x
x
xxx
![Page 16: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/16.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
146325)( 712
29
xxxxxf
)(lim xfx
![Page 17: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/17.jpg)
To show that, we follow the following steps:
0)(lim
xfx
060
0)0(46)0(3)0(2)0(5
1lim1lim46lim
1lim31lim21lim5
1146lim
3125lim
1146
3125
lim146325lim)(lim
125
12103
125
12103
125
12103
712
29
xx
xxx
xx
xxx
xx
xxxxxxxxf
xxx
xxx
x
x
xxx
![Page 18: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/18.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
146325)( 79
212
xxxxxf
)(lim xfx
![Page 19: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/19.jpg)
They are infinite limits. To show that, we follow the following steps:
3
9
12
79
212
lim65lim
146325lim)(lim
xassamethearewhichxxassametheare
xxxxxf
x
x
xx
.)(lim existnotdoxfx
![Page 20: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/20.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
146325)( 79
212
xxxxxf
)(lim xfx
![Page 21: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/21.jpg)
They are infinite limits. To show that, we follow the following steps:
)(lim65lim
146325lim)(lim
3
9
12
79
212
xassamethearewhichxxassametheare
xxxxxf
x
x
xx
.)(lim existnotdoxfx
![Page 22: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/22.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
146325)( 78
212
xxxxxf
)(lim xfx
![Page 23: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/23.jpg)
They are infinite limits. To show that, we follow the following steps:
.)(lim existnotdoxfx
4
8
12
78
212
lim65lim
146325lim)(lim
xassamethearewhichxxassametheare
xxxxxf
x
x
xx
![Page 24: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/24.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
146325)( 78
212
xxxxxf
)(lim xfx
![Page 25: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/25.jpg)
They are infinite limits. To show that, we follow the following steps:
.)(lim existnotdoxfx
)(lim65lim
146325lim)(lim
4
8
12
78
212
xassamethearewhichxxassametheare
xxxxxf
x
x
xx
![Page 26: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/26.jpg)
Limits @ Infinity
2. Problems Involving Roots
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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) = 2
![Page 28: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/28.jpg)
Example
0;16
0;30;16
9
92
92
2
2
29
29
2
2
2
16
16
16
916)(
:916)(
:
xx
xxx
x
x
x
x
x
x
x
x
xxf
asrewwrittenbecanxxf
Let
![Page 29: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/29.jpg)
Example (1)
fofasymptoteshorizontaltheareyandylinestheThusWhy
x
x
xx
xx
xxxf
atandatfofitsmlithefrstfindWeasymptoteshorizontalFinding
Solution
fofasymptotesallFindxxxf
Let
xxxx
22?2
)22(
916lim
)22(
916lim
22916lim)(lim
:.1:
22916)(
:
222
2
![Page 30: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/30.jpg)
)1(2916lim)(lim
,.015916,1
:1,)1(2916lim)(lim
,.015916,1
1
:.122916)(
2
11
2
2
11
2
2
xxxf
Thusnegativekeepingwhileapproachesxandapproachesxleftthefromapproachesxas
asymptoteverticalaisxhencexxxf
Thuspositivekeepingwhileapproachesxandapproachesxrightthefromapproachesxas
numeratorthenotandnatordomitheofzeroaisxmitlifinitenianhasfwherefindfirstWe
asymptotesverticalFindingxxxf
xx
xx
![Page 31: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/31.jpg)
)23,0(int
23
2090)0(
potheataxisythewithectionInters
f
![Page 32: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/32.jpg)
Example (2)
fofasymptoteshorizontaltheareyandylinestheThus
x
x
xx
xx
xxxf
exampleprevioustheindidweasatandatfofitsmlithefindfrstWeasymptoteshorizontalFinding
Solution
itgraphfofasymptotesallFindxxxf
Let
xxxx
222
)22(
916lim
)22(
916lim
22916lim)(lim
)(:.1
:
!&22916)(
:
222
2
![Page 33: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/33.jpg)
)1(2916lim)(lim
,.015916,1
:1,)1(2916lim)(lim
,.015916,1
1
:.122916)(
2
11
2
2
11
2
2
xxxf
Thusnegativekeepingwhileapproachesxandapproachesxleftthefromxapproachesxas
asymptoteverticalaisxhencexxxf
Thuspositivekeepingwhileapproachesxandapproachesxrightthefromapproachesxas
numeratorthenotandnatordomitheofzeroaisxmitlifinitenianhasfwherefindfirstWe
asymptotesverticalFindingxxxf
xx
xx
![Page 34: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/34.jpg)
![Page 35: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/35.jpg)
Homework:Problems:Example (4) & (5) –Section 3.4. Page: 228Exercises 3.4 Page: 235 :Problems: 9, 10, 11, 15, 23, 25,17 & 40
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4. The case of infinite limit
See the examples in the file on the infinite limits and also the examples of infinite limits in the file on limits at infinity.
![Page 37: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/37.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 38: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/38.jpg)
Infinite Limits- The Case of Rational Functions
A rational function has an infinite limit if the limit of the denominator is zero and the limit 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.
![Page 39: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/39.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 40: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/40.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 41: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/41.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
15)(
xxxf
![Page 42: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/42.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 43: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/43.jpg)
Example (3)Let
Find Solution:First, rewrite: x=0 is a zero of the denominator which is not a zero of the numerator.
)(lim.0
xfax
)(lim.0
xfbx
54
11)(xx
xf
554
111)(xx
xxxf
![Page 44: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/44.jpg)
)(lim0
xfx
The function has a vertical asymptote at x = 0, which is the line x = 0
)(lim0
xfx
a. As x approaches 0 from the right, the numerator approaches -1 (thus keeping negative), and the denominator approaches 0 while keeping positive; hence, the function decreases with no bound. Thus:
b. As x approaches 0 from the left, the numerator approaches -1 (thus keeping negative), and the denominator approaches 0 while keeping negative; hence, the function increases with no bound. Thus:
![Page 45: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/45.jpg)
Same Type Problems from the Homework
Exercises 1.5 Pages 59-61Problems: 29, 31, 33, 37
![Page 46: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/46.jpg)
5. The case of discontinuous function f defined by a formula involving absolute values
fGraphQuestion
existnotdoesxx
xx
xx
xx
xffunctionThe
Solutionxx
existsifFindExample
x
xx
xx
xx
xxx
x
:5153
lim
5153
lim335153
lim
5153
)(
:5153
lim
:,:)1(
5
55
5;35;3
5;5153
5;5
)153(
5
![Page 47: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/47.jpg)
5. The case of discontinuous function f defined by a formula involving absolute values
existnotdoesxx
xfx
xf
xxxffunctionThe
Solution
xx
existsifFindExample
x
xxxx
xx
x
xxx
xxx
x
21
21lim
00lim)(lim,1lim)(lim
21
21)(
:
21
21lim
:,:)2(
0
0000
0;1
0;0
0;21
21
0;21
21
0
![Page 48: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/48.jpg)
Same Type Problems from the Homework
Exercises 1.6 Pages 69-71Problems: 43, 45, 42, 44
![Page 49: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/49.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
)()(limxhxg
ax
)(lim&)(lim xhRcxgaxax
0)()(lim
xhxg
ax
![Page 50: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/50.jpg)
Example on the case constant/∞
011lim
)()(lim
,
?)()(lim1)(
11lim)(lim1)(::11lim
,:)1(
xxxhxg
Thus
Whyxhxxxh
xgxghaveWe
Solutionxx
FindExample
xx
x
xx
x
![Page 51: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/51.jpg)
Same Type Problems
36
24
5
23lim.2
1
25lim.1
,:)1(
xx
xxx
FindExample
x
x
![Page 52: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/52.jpg)
7. Other cases: the case ∞- ∞, 0∙∞
Suppose we want to find:
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.
arebothorarehandgfunctionsbothofmitslithewhen
xhxgOrxhxgxax
)()(lim)()(lim
![Page 53: Algebra of Limits](https://reader035.vdocument.in/reader035/viewer/2022062410/568161e5550346895dd20539/html5/thumbnails/53.jpg)
Example for the case ∞- ∞
casethehaveweThus
xxhxxh
xxxxg
xx
xxxxg
Discussion
xx
existsifFindExample
xx
xxx
xx
x
x
xx
x
x
,
lim)(lim)(
71lim7lim)(lim
71717)(
:
7lim
:,:)1(
22
0;71
0;7
0;71
2222
2
2
2
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?07
7lim
7
)7(lim
7
77lim7lim
2
2
22
2
222
Whyxx
xx
xxxx
xxxxxx
x
x
xx
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121
12
1lim)2)(1(
1lim
0/0,23
1lim231)2(lim
)2)(1(1
11lim
231
11lim
,23
1lim&1
1lim
231lim&
11lim
::
231
11lim
:,:)2(
11
2121
121
211
211
21
xxxx
casethehaveweNowxx
xxx
x
xxxxxx
casethehaveweThusxxx
xxx
haveWeSolution
xxx
existsifFindExample
xx
xx
xx
xx
xx
x
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)21:(
12
11lim.1
)21:(1
11lim.1
:,:
21
0
Answerxx
Answerxxx
existifFindPeoblems
x
x
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7. Using the Squeeze Theorem
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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 that in this case the limit of f at x=a will equal the limit of g and h at this point.
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The Squeeze Theorem
lxfThen
xhlxg
dcawhereadcxxhxfxg
Let
ax
axax
)(:
)()(&
),(),(;)()()(
:
lim
limlim
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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
xftheoremsqueezethebyThus
xxxfx
xx
haveWeSoluion
xfFind
xxxfx
Let
x
xx
x
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Example (2)
7)(,,
),0[;)(12,),0(2
&
771616)74(
,7916)94(::
)(
),0[;74)(94
:
lim
limlim
lim
4
2
2
4
4
4
2
xftheoremsqueezethebyThus
xxxfx
xx
xhaveWe
Soluion
xfFind
xxxxfx
Let
x
x
x
x
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Example (3)
0)1sin(
,
)(0)(
'
0)5,5(;1sin
)...(
?0)5,5(;11sin1
::
:
)1sin(
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
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Example (4)
0)2cos(
,
)(0)(
'
0)5,5(;2cos
)...(
?0)5,5(;12cos1
::
:
)2cos(
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
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Same Type Problems from the Homework
Exercises 6.1 Pages 70Problems: 37, 38, 39