c# pdf free : Add hyperlink to pdf control software system azure html web page console TRENCH_REAL_ANALYSIS4-part259

Section2.1
FunctionsandLimits
33
and
.f
1
f
2
f
n
/.x/Df
1
.x/f
2
.x/f
n
.x/
(2.1.3)
onD D
T
n
iD1
D
f
i
,providedthatDisnonempty. Iff
1
D f
2
DDf
n
,then(2.1.3)
definesthenthpoweroff:
.f
n
/.x/D.f.x//
n
:
Fromthesedefinitions,wecanbuildthesetofallpolynomials
p.x/Da
0
Ca
1
xCCa
n
x
n
;
startingfromtheconstantfunctionsandf.x/Dx. Thequotientoftwopolynomialsisa
rationalfunction
r.x/D
a
0
Ca
1
xCCa
n
x
n
b
0
Cb
1
xCCb
m
xm
.b
m
¤0/:
Thedomainofristhesetofpointswherethedenominatorisnonzero.
Limits
Theessenceoftheconceptoflimitforreal-valuedfunctionsofarealvariableisthis:IfL
isarealnumber,thenlim
x!x
0
f.x/DLmeansthatthevaluef.x/canbemadeasclose
toLaswewishbytakingxsufficientlyclosetox
0
.Thisismadepreciseinthefollowing
definition.
y
x
L + 
L − 
L
y = f(x)
x0 − δ 
x0 + δ 
x0 
Figure2.1.1
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34 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Definition2.1.2
Wesaythatf.x/approachesthelimitLasxapproachesx
0
,and
write
lim
x!x
0
f.x/DL;
iff isdefinedonsomedeletedneighborhoodofx
0
and,forevery>0,thereisaı>0
suchthat
jf.x/Lj<
(2.1.4)
if
0<jxx
0
j<ı:
(2.1.5)
Figure2.1.1depictsthegraphofafunctionforwhichlim
x!x
0
f.x/exists.
Example2.1.5
Ifcandxarearbitraryrealnumbersandf.x/Dcx,then
lim
x!x
0
f.x/Dcx
0
:
Toprovethis,wewrite
jf.x/cx
0
jDjcxcx
0
jDjcjjxx
0
j:
Ifc¤0,thisyields
jf.x/cx
0
j<
(2.1.6)
if
jxx
0
j<ı;
whereıisanynumbersuchthat0<ı=jcj.IfcD0,thenf.x/cx
0
D0forallx,
so(2.1.6)holdsforallx.
WeemphasizethatDefinition2.1.2doesnotinvolvef.x
0
/,orevenrequirethatitbe
defined,since(2.1.5)excludesthecasewherexDx
0
.
Example2.1.6
If
f.x/Dxsin
1
x
; x¤0;
then
lim
x!0
f.x/D0
eventhoughf isnotdefinedatx
0
D0,becauseif
0<jxj<ıD;
then
jf.x/0jD
ˇ
ˇ
ˇ
ˇ
xsin
1
x
ˇ
ˇ
ˇ
ˇ
jxj<:
Ontheotherhand,thefunction
g.x/Dsin
1
x
; x¤0;
has nolimitas x approaches0, sinceitassumesallvaluesbetween1and1inevery
neighborhoodoftheorigin(Exercise2.1.26).
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Section2.1
FunctionsandLimits
35
Thenexttheoremsaysthatafunctioncannothavemorethanonelimitatapoint.
Theorem2.1.3
Iflim
x!x
0
f.x/exists;thenitisuniqueIthatis;if
lim
x!x
0
f.x/DL
1
and
lim
x!x
0
f.x/DL
2
;
(2.1.7)
thenL
1
DL
2
:
Proof
Supposethat(2.1.7)holdsandlet>0.FromDefinition2.1.2,therearepositive
numbersı
1
andı
2
suchthat
jf.x/L
i
j< if 0<jxx
0
j<ı
i
; i i D1;2:
IfıDmin.ı
1
2
/,then
jL
1
L
2
jDjL
1
f.x/Cf.x/L
2
j
jL
1
f.x/jCjf.x/L
2
j<2 if 0<jxx
0
j<ı:
Wehavenowestablishedaninequalitythatdoesnotdependonx;thatis,
jL
1
L
2
j<2:
Sincethisholdsforanypositive,L
1
DL
2
.
Definition2.1.2isnotchangedbyreplacing(2.1.4)with
jf.x/Lj<K;
(2.1.8)
whereKisapositiveconstant,becauseifeitherof(2.1.4)or(2.1.8)canbemadetohold
forany >0bymakingjxx
0
jsufficientlysmallandpositive,thensocantheother
(Exercise2.1.5).Thismayseemtobeaminorpoint,butitisoftenconvenienttoworkwith
(2.1.8)ratherthan(2.1.4),aswewillseeintheproofofthefollowingtheorem.
AUsefulTheoremaboutLimits
Theorem2.1.4
If
lim
x!x
0
f.x/DL
1
and
lim
x!x
0
g.x/DL
2
;
(2.1.9)
then
lim
x!x
0
.f Cg/.x/DL
1
CL
2
;
(2.1.10)
lim
x!x
0
.fg/.x/DL
1
L
2
;
(2.1.11)
lim
x!x
0
.fg/.x/DL
1
L
2
;
(2.1.12)
and,ifL
2
¤0,
(2.1.13)
lim
x!x
0
f
g
.x/D
L
1
L
2
:
(2.1.14)
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36 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Proof
From(2.1.9)andDefinition2.1.2,if>0,thereisaı
1
>0suchthat
jf.x/L
1
j<
(2.1.15)
if0<jxx
0
j<ı
1
,andaı
2
>0suchthat
jg.x/L
2
j<
(2.1.16)
if0<jxx
0
j<ı
2
.Supposethat
0<jxx
0
j<ıDmin.ı
1
2
/;
(2.1.17)
sothat(2.1.15)and(2.1.16)bothhold.Then
j.f ˙g/.x/.L
1
˙L
2
/jDj.f.x/L
1
/˙.g.x/L
2
/j
jf.x/L
1
jCjg.x/L
2
j<2;
whichproves(2.1.10)and(2.1.11).
Toprove(2.1.12),weassume(2.1.17)andwrite
j.fg/.x/L
1
L
2
jDjf.x/g.x/L
1
L
2
j
Djf.x/.g.x/L
2
/CL
2
.f.x/L
1
/j
jf.x/jjg.x/L
2
jCjL
2
jjf.x/L
1
j
.jf.x/jCjL
2
j/ (from(2.1.15)and(2.1.16))
.jf.x/L
1
jCjL
1
jCjL
2
j/
.CjL
1
jCjL
2
j/ from(2.1.15)
.1CjL
1
jCjL
2
j/
if<1andxsatisfies(2.1.17).Thisproves(2.1.12).
Toprove(2.1.14),wefirstobservethatifL
2
¤0,thereisaı
3
>0suchthat
jg.x/L
2
j<
jL
2
j
2
;
so
jg.x/j>
jL
2
j
2
(2.1.18)
if
0<jxx
0
j<ı
3
:
Toseethis,letLDL
2
andDjL
2
j=2in(2.1.4).Nowsupposethat
0<jxx
0
j<min.ı
1
2
3
/;
sothat(2.1.15),(2.1.16),and(2.1.18)allhold.Then
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Section2.1
FunctionsandLimits
37
ˇ
ˇ
ˇ
ˇ
f
g
.x/
L
1
L
2
ˇ
ˇ
ˇ
ˇ
D
ˇ
ˇ
ˇ
ˇ
f.x/
g.x/
L
1
L
2
ˇ
ˇ
ˇ
ˇ
D
jL
2
f.x/L
1
g.x/j
jg.x/L
2
j
2
jL
2
j2
jL
2
f.x/L
1
g.x/j
D
2
jL
2
j2
jL
2
Œf.x/L
1
CL
1
ŒL
2
g.x/j (from(2.1.18))
2
jL
2
j2
ŒjL
2
jjf.x/L
1
jCjL
1
jjL
2
g.x/j
2
jL
2
j2
.jL
2
jCjL
1
j/ (from(2.1.15)and(2.1.16)):
Thisproves(2.1.14).
SuccessiveapplicationsofthevariouspartsofTheorem2.1.4permitustofindlimits
withoutthe–ıargumentsrequiredbyDefinition2.1.2.
Example2.1.7
UseTheorem2.1.4tofind
lim
x!2
9x2
xC1
and
lim
x!2
.9x
2
/.xC1/:
Solution
Ifcisaconstant,thenlim
x!x
0
cDc,and,fromExample2.1.5,lim
x!x
0
xD
x
0
.Therefore,fromTheorem2.1.4,
lim
x!2
.9x
2
/D lim
x!2
9lim
x!2
x
2
D lim
x!2
9.lim
x!2
x/
2
D92
2
D5;
and
lim
x!2
.xC1/D lim
x!2
xC lim
x!2
1D2C1D3:
Therefore,
lim
x!2
9x
2
xC1
D
lim
x!2
.9x
2
/
lim
x!2
.xC1/
D
5
3
and
lim
x!2
.9x
2
/.xC1/D lim
x!2
.9x
2
/lim
x!2
.xC1/D53D15:
One-SidedLimits
Thefunction
f.x/D2x sin
p
x
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38 Chapter2
DifferentialCalculusofFunctionsofOneVariable
satisfiestheinequality
jf.x/j<
if0 < x x < ı ı D =2. However,thisdoesnotmeanthatlim
x!0
f.x/ D 0,sincef is
notdefinedfornegativex,asitmustbetosatisfytheconditionsofDefinition2.1.2with
x
0
D0andLD0.Thefunction
g.x/DxC
jxj
x
; x¤0;
canberewrittenas
g.x/D
xC1; x>0;
x1; x<0I
hence,everyopenintervalcontainingx
0
D 0alsocontainspointsx
1
andx
2
suchthat
jg.x
1
/g.x
2
/jisas closeto2asweplease. Therefore,lim
x!x
0
g.x/doesnotexist
(Exercise2.1.26).
Althoughf.x/andg.x/donotapproachlimitsasxapproacheszero,theyeachexhibit
adefinitesortoflimitingbehaviorforsmallpositivevaluesofx,asdoesg.x/forsmall
negativevaluesofx.Thekindofbehaviorwehaveinmindisdefinedpreciselyasfollows.
y
x
x
0
x    x
0
x    x
0
+
f(x) = λ
y = f(x)
f(x) = µ
lim
lim
µ
λ
Figure2.1.2
Definition2.1.5
(a)
Wesaythatf.x/approachestheleft-handlimitLasxapproachesx
0
fromtheleft,
andwrite
lim
x!x
0
f.x/DL;
iff isdefinedonsomeopeninterval.a;x
0
/and,foreach >0,thereisaı>0
suchthat
jf.x/Lj< if x
0
ı<x<x
0
:
Section2.1
FunctionsandLimits
39
(b)
Wesaythatf.x/approachestheright-handlimitLasx approachesx
0
fromthe
right,andwrite
lim
x!x
0
C
f.x/DL;
iff isdefinedonsomeopeninterval.x
0
;b/and,foreach >0,thereisaı>0
suchthat
jf.x/Lj< if x
0
<x<x
0
Cı:
Figure2.1.2showsthegraphofafunctionthathasdistinctleft-andright-handlimitsat
apointx
0
.
Example2.1.8
Let
f.x/D
x
jxj
; x¤0:
Ifx<0,thenf.x/Dx=xD1,so
lim
x!0
f.x/D1:
Ifx>0,thenf.x/Dx=xD1,so
lim
x!0C
f.x/D1:
Example2.1.9
Let
g.x/D
xCjxj.1Cx/
x
sin
1
x
; x¤0:
Ifx<0,then
g.x/Dxsin
1
x
;
so
lim
x!0
g.x/D0;
since
jg.x/0jD
ˇ
ˇ
ˇ
ˇ
xsin
1
x
ˇ
ˇ
ˇ
ˇ
jxj<
if<x<0;thatis,Definition2.1.5
(a)
issatisfiedwithıD.Ifx>0,then
g.x/D.2Cx/sin
1
x
;
whichtakesoneveryvaluebetween2and2ineveryinterval.0;ı/.Hence,g.x/doesnot
approacharight-handlimitatxapproaches0fromtheright. Thisshowsthatafunction
mayhavealimitfromonesideatapointbutfailtohavealimitfromtheotherside.
40 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Example2.1.10
Weleaveittoyoutoverifythat
lim
x!0C
jxj
x
Cx
D
1;
lim
x!0
jxj
x
Cx
D1;
lim
x!0C
xsin
p
xD
0;
andlim
x!0
sin
p
xdoesnotexist.
Left-andright-handlimitsarealsocalledone-sidedlimits. Wewilloftensimplifythe
notationbywriting
lim
x!x
0
f.x/Df.x
0
/ and
lim
x!x
0
C
f.x/Df.x
0
C/:
Thefollowingtheoremstatestheconnectionbetweenlimitsandone-sidedlimits. We
leavetheprooftoyou(Exercise2.1.12).
Theorem2.1.6
Afunctionfhasalimitatx
0
ifandonlyifithasleft-andright-hand
limitsatx
0
;andtheyareequal.Morespecifically;
lim
x!x
0
f.x/DL
ifandonlyif
f.x
0
C/Df.x
0
/DL:
Withonlyminormodificationsoftheirproofs(replacingtheinequality0<jxx
0
j<ı
byx
0
ı<x <x
0
orx
0
<x <x
0
Cı),itcanbeshownthattheassertionsofTheo-
rems2.1.3and2.1.4remainvalidif“lim
x!x
0
”isreplacedby“lim
x!x
0
”or“lim
x!x
0
C
throughout(Exercise2.1.13).
Limitsat
˙1
Limitsandone-sidedlimitshavetodowiththebehaviorofafunctionf nearalimitpoint
ofD
f
.ItisequallyreasonabletostudyfforlargepositivevaluesofxifD
f
isunbounded
aboveorforlargenegativevaluesofxifD
f
isunboundedbelow.
Definition2.1.7
Wesaythatf.x/approachesthelimitLasx approaches1,and
write
lim
x!1
f.x/DL;
iffisdefinedonaninterval.a;1/and,foreach>0,thereisanumberˇsuchthat
jf.x/Lj<
if x>ˇ:
Section2.1
FunctionsandLimits
41
Figure2.1.3providesanillustrationofthesituationdescribedinDefinition2.1.7.
x    ∞
lim
f(x) = L
β
y
L +
L −
L
x
Figure2.1.3
Weleaveittoyoutodefinethestatement“lim
x!1
f.x/DL”(Exercise2.1.14)and
toshowthatTheorems2.1.3and2.1.4remainvalidifx
0
isreplacedthroughoutby1or
1(Exercise2.1.16).
Example2.1.11
Let
f.x/D1
1
x2
; g.x/D
2jxj
1Cx
; and h.x/Dsinx:
Then
lim
x!1
f.x/D1;
since
jf.x/1jD
1
x2
< if x>
1
p
;
and
lim
x!1
g.x/D2;
since
jg.x/2jD
ˇ
ˇ
ˇ
ˇ
2x
1Cx
2
ˇ
ˇ
ˇ
ˇ
D
2
1Cx
<
2
x
< if x>
2
:
However,lim
x!1
h.x/doesnotexist,sincehassumesallvaluesbetween1and1inany
semi-infiniteinterval.;1/.
Weleave ittoyouto showthat lim
x!1
f.x/ D 1, , lim
x!1
g.x/ D 2, , and
lim
x!1
h.x/doesnotexist(Exercise2.1.17).
42 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Wewillsometimes denotelim
x!1
f.x/andlim
x!1
f.x/byf.1/andf.1/,
respectively.
Infinite Limits
Thefunctions
f.x/D
1
x
; g.x/D
1
x2
; p.x/Dsin
1
x
;
and
q.x/D
1
x2
sin
1
x
donothavelimits,orevenone-sidedlimits,atx
0
D0.Theyfailtohavelimitsindifferent
ways:
 f.x/increasesbeyondboundasxapproaches0fromtherightanddecreasesbeyond
boundasxapproaches0fromtheleft;
 g.x/increasesbeyondboundasxapproacheszero;
 p.x/oscillateswithever-increasingfrequencyasxapproacheszero;
 q.x/oscillateswithever-increasingamplitudeandfrequencyasxapproaches0.
Thekindofbehaviorexhibitedbyf andgnearx
0
D 0issufficientlycommonand
simpletoleadustodefineinfinitelimits.
Definition2.1.8
Wesaythatf.x/approaches1asxapproachesx
0
fromtheleft,
andwrite
lim
x!x
0
f.x/D1
or f.x
0
/D1;
iff isdefinedonaninterval.a;x
0
/and,foreachrealnumberM,thereisaı>0such
that
f.x/>M
if x
0
ı<x<x
0
:
Example2.1.12
Weleaveittoyoutodefinetheotherkindsofinfinitelimits(Exer-
cises2.1.19and2.1.21)andshowthat
lim
x!0
1
x
D1;
lim
x!0C
1
x
D1I
lim
x!0
1
x2
D lim
x!0C
1
x2
D lim
x!0
1
x2
D1I
lim
x!1
x
2
D lim
x!1
x
2
D1I
and
lim
x!1
x
3
D1;
lim
x!1
x
3
D1:
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