c# pdf free : Clickable links in pdf software control project winforms azure .net UWP TRENCH_REAL_ANALYSIS10-part227

Section2.4
L’Hospital’sRule
93
Therefore,lim
x!0
f
0
.x/=g
0
.x/doesnotexist.However,
lim
x!0
f.x/
g.x/
D lim
x!0
1xsin.1=x/
.sinx/=x
D
1
1
D1:
TheIndeterminateForm
01
Wesaythataproductfgisoftheform01asx!bifoneofthefactorsapproaches
0andtheotherapproaches ˙1as x ! b. Inthiscase, , itmaybeusefultoapply
L’Hospital’sruleafterwriting
f.x/g.x/D
f.x/
1=g.x/
or f.x/g.x/D
g.x/
1=f.x/
;
sinceoneoftheseratiosisoftheform0=0andtheotherisoftheform1=1asx!b.
Similarstatementsapplytolimitsasx!bC,x!b,andx!˙1.
Example2.4.6
Theproductxlogxisoftheform01asx!0C.Convertingitto
an1=1formyields
lim
x!0C
xlogxD lim
x!0C
logx
1=x
D lim
x!0C
1=x
1=x2
D lim
x!0C
xD0:
Convertingtoa0=0formleadstoamorecomplicatedproblem:
lim
x!0C
xlogxD lim
x!0C
x
1=logx
D lim
x!0C
1
1=x.logx/2
D lim
x!0C
x.logx/
2
D‹
Example2.4.7
Theproductxlog.1C1=x/isoftheform01asx!1.Converting
ittoa0=0formyields
lim
x!1
xlog.1C1=x/D lim
x!1
log.1C1=x/
1=x
D lim
x!1
Œ1=.1C1=x/.1=x
2
/
1=x2
D lim
x!1
1
1C1=x
D1:
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94 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Inthiscase,convertingtoan1=1formcomplicatestheproblem:
lim
x!1
xlog.1C1=x/D lim
x!1
x
1=log.1C1=x/
D lim
x!1
1
1
Œlog.1C1=x/2

1=x
2
1C1=x
D lim
x!1
x.xC1/Œlog.1C1=x/
2
D‹
TheIndeterminateForm
11
Adifferencefgisoftheform11asx!bif
lim
x!b
f.x/D lim
x!b
g.x/D˙1:
Inthiscase,itmaybepossibletomanipulatef gintoanexpressionthatisnolonger
indeterminate,orisoftheform0=0or1=1asx!b.Similarremarksapplytolimits
asx!bC,x!b,orx!˙1.
Example2.4.8
Thedifference
sinx
x2
1
x
isoftheform11asx!0,butitcanberewrittenasthe0=0form
sinxx
x2
:
Hence,
lim
x!0
sinx
x2
1
x
D lim
x!0
sinxx
x2
D lim
x!0
cosx1
2x
D lim
x!0
sinx
2
D0:
Example2.4.9
Thedifference
x
2
x
isoftheform11asx!1.Rewritingitas
x
2
1
1
x
;
whichisnolongerindeterminateasx!1,wefindthat
lim
x!1
.x
2
x/D lim
x!1
x
2
1
1
x
D
lim
x!1
x
2
lim
x!1
1
1
x
D.1/.1/D1
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Section2.4
L’Hospital’sRule
95
TheIndeterminateForms
0
0
,
1
1
,and
1
0
Thefunctionfisdefinedby
f.x/
g.x/
De
g.x/logf.x/
Dexp.g.x/logf.x//
forallxsuchthatf.x/>0.Therefore,iffandgaredefinedandf.x/>0onaninterval
.a;b/,Exercise2.2.22impliesthat
lim
x!b
Œf.x/
g.x/
Dexp
lim
x!b
g.x/logf.x/
(2.4.14)
iflim
x!b
g.x/logf.x/existsintheextendedreals.(Ifthislimitis˙1then(2.4.14)is
validifwedefinee
1
D0ande
1
D1.)Theproductglogf canbeoftheform01
inthreewaysasx!b:
(a)
Iflim
x!b
g.x/D0andlim
x!b
f.x/D0.
(b)
Iflim
x!b
g.x/D˙1andlim
x!b
f.x/D1.
(c)
Iflim
x!b
g.x/D0andlim
x!b
f.x/D1.
Inthesethreecases,wesaythatf
g
isoftheform0
0
,1
1
,and1
0
,respectively,asx!
b.Similardefinitionsapplytolimitsasx!bC,x!b,andx!˙1.
Example2.4.10
Thefunctionx
x
isoftheform0
0
asx!0C.Since
x
x
De
xlogx
andlim
x!0C
xlogxD0(Example2.4.6),
lim
x!0C
x
x
De
0
D1:
Example2.4.11
Thefunctionx
1=.x1/
isoftheform1
1
asx!1.Since
x
1=.x1/
Dexp
logx
x1
and
lim
x!1
logx
x1
D lim
x!1
1=x
1
D1;
itfollowsthat
lim
x!1
x
1=.x1/
De
1
De:
Example2.4.12
Thefunctionx1=x isoftheform10asx!1.Since
x
1=x
Dexp
logx
x
and
lim
x!1
logx
x
D lim
x!1
1=x
1
D0;
96 Chapter2
DifferentialCalculusofFunctionsofOneVariable
itfollowsthat
lim
x!1
x
1=x
De
0
D1:
2.4Exercises
1.
ProveTheorem2.4.1forthecasewherelim
x!b
f
0
.x/=g
0
.x/D˙1.
InExercises2.4.22.4.40,findtheindicatedlimits.
2.
lim
x!0
tan1x
sin
1
x
3.
lim
x!0
1cosx
log.1Cx2/
4.
lim
x!0C
1Ccosx
e1
5.
lim
x!
sinnx
sinx
6.
lim
x!0
log.1Cx/
x
7.
lim
x!1
e
x
sine
x
2
8.
lim
x!1
xsin.1=x/
9.
lim
x!1
p
x.e
1=x
1/
10.
lim
x!0C
tanxlogx
11.
lim
x!
sinxlog.jtanxj/
12.
lim
x!0C
1
x
Clog.tanx/
13.
lim
x!1
.
p
xC1
p
x/
14.
lim
x!0
1
e1
1
x
15.
lim
x!0
.cotxcscx/
16.
lim
x!0
1
sinx
1
x
17.
lim
x!
jsinxj
tanx
18.
lim
x!=2
jtanxj
cosx
19.
lim
x!0
jsinxj
x
20.
lim
x!0
.1Cx/
1=x
21.
lim
x!1
x
sin.1=x/
22.
lim
x!0
x
1cosx
2
x
23.
lim
x!0C
x
˛
logx
24.
lim
x!e
log.logx/
sin.xe/
25.
lim
x!1
xC1
x1
p
x
2
1
26.
lim
x!1C
xC1
x1
p
x
2
1
27.
lim
x!1
.logx/
ˇ
x
28.
lim
x!1
.coshxsinhx/
29.
lim
x!1
.x
˛
logx/
30.
lim
x!1
e
x
2
sin.e
x
/
Section2.4
L’Hospital’sRule
97
31.
lim
x!1
x.xC1/Œlog.1C1=x/
2
32.
lim
x!0
sinxxCx
3
=6
x5
33.
lim
x!1
e
x
x˛
34.
lim
x!3=2
e
tanx
cosx
35.
lim
x!1C
.logx/
˛
log.logx/
36.
lim
x!1
x
x
xlogx
37.
lim
x!=2
.sinx/
tanx
38.
lim
x!0
e
x
Xn
rD0
x
r
xn
.nD integer1/
39.
lim
x!0
sinx
Xn
rD0
.1/
r
x
2rC1
.2rC1/Š
x2nC1
.nD integer0/
40.
lim
x!0
e
1=x
2
xn
D0 (nDinteger)
41. (a)
Prove:Iff iscontinuousatx
0
andlim
x!x
0
f
0
.x/exists,thenf
0
.x
0
/exists
andf
0
iscontinuousatx
0
.
(b)
Giveanexampletoshowthatitisnecessarytoassumein
(a)
thatf iscon-
tinuousatx
0
.
42.
TheiteratedlogarithmsaredefinedbyL
0
.x/Dxand
L
n
.x/Dlog.L
n1
.x//; x>a
n
; n1;
wherea
1
D0anda
n
De
a
n1
;n1.Showthat
(a)
L
n
.x/DL
n1
.logx/; x>a
n
; n1.
(b)
L
n1
.a
n
C/D0andL
n
.a
n
C/D1.
(c)
lim
x!a
n
C
.L
n1
.x//
˛
L
n
.x/D0if˛>0andn1.
(d)
lim
x!1
.L
n
.x//
˛
=L
n1
.x/D0if˛isarbitraryandn1.
43.
Letfbepositiveanddifferentiableon.0;1/,andsupposethat
lim
x!1
f
0
.x/
f.x/
DL; where 0<L1:
Definef
0
.x/Dxand
f
n
.x/Df.f
n1
.x//; n1:
UseL’Hospital’sruletoshowthat
lim
x!1
.f
n
.x//
˛
f
n1
.x/
D1
if ˛>0 and n1:
98 Chapter2
DifferentialCalculusofFunctionsofOneVariable
44.
Letf bedifferentiableonsomedeletedneighborhoodN N ofx
0
,andsupposethatf
andf
0
havenozerosinN.Find
(a)
lim
x!x
0
jf.x/j
f.x/
if lim
x!x
0
f.x/D0;
(b)
lim
x!x
0
jf.x/j
1=.f.x/1/
if lim
x!x
0
f.x/D1;
(c)
lim
x!x
0
jf.x/j
1=f.x/
iflim
x!x
0
f.x/D1.
45.
Supposethatf andgaredifferentiableandg
0
hasnozeroson.a;b/.Supposealso
thatlim
x!b
f
0
.x/=g
0
.x/DLandeither
lim
x!b
f.x/D lim
x!b
g.x/D0
or
lim
x!b
f.x/D1
and
lim
x!b
g.x/D˙1:
Findlim
x!b
.1Cf.x//
1=g.x/
.
46.
Wedistinguishbetween11.D1/and.1/1.D1/andbetween1C1
.D 1/and11.D 1/. Whydon’twedistinguishbetween01and
0.1/,11and1C1,1=1and1=1,and1
1
and1
1
?
2.5TAYLOR’STHEOREM
Apolynomialisafunctionoftheform
p.x/Da
0
Ca
1
.xx
0
/CCa
n
.xx
0
/
n
;
(2.5.1)
wherea
0
,...,a
n
andx
0
areconstants.Sinceitiseasytocalculatethevaluesofapolyno-
mial,considerableefforthasbeendevotedtousingthemtoapproximatemorecomplicated
functions.Taylor’stheoremisoneoftheoldestandmostimportantresultsonthisquestion.
Thepolynomial(2.5.1)issaidtobewritteninpowersofxx
0
,andisofdegreenif
a
n
¤0. Ifwewishtoleaveopenthepossibilitythata
n
D0,wesaythatpisofdegree
 n. Inparticular, , aconstantpolynomialp.x/ D D a
0
isofdegreezeroifa
0
¤ 0. If
a
0
D0,sothatpvanishesidentically,thenphasnodegreeaccordingtoourdefinition,
whichrequiresatleastonecoefficienttobenonzero. Forconveniencewesaythatthe
identicallyzeropolynomialphasdegree1.(Anynegativenumberwoulddoaswellas
1.Thepointisthatwiththisconvention,thestatementthatpisapolynomialofdegree
nincludesthepossibilitythatpisidenticallyzero.)
TaylorPolynomials
WesawinLemma2.3.2thatiffisdifferentiableatx
0
,then
f.x/Df.x
0
/Cf
0
.x
0
/.xx
0
/CE.x/.xx
0
/;
Section2.5
Taylor’sTheorem
99
where
lim
x!x
0
E.x/D0:
Togeneralizethisresult,wefirstrestateit:thepolynomial
T
1
.x/Df.x
0
/Cf
0
.x
0
/.xx
0
/;
whichisofdegree1andsatisfies
T
1
.x
0
/Df.x
0
/; T
0
1
.x
0
/Df
0
.x
0
/;
approximatesf sowellnearx
0
that
lim
x!x
0
f.x/T
1
.x/
xx
0
D0:
(2.5.2)
Nowsupposethatf hasnderivativesatx
0
andT
n
isthepolynomialofdegree n
suchthat
T
.r/
n
.x
0
/Df
.r/
.x
0
/; 0rn:
(2.5.3)
HowwelldoesT
n
approximatef nearx
0
?
Toanswerthisquestion,wemustfirstfindT
n
.SinceT
n
isapolynomialofdegree n,
itcanbewrittenas
T
n
.x/Da
0
Ca
1
.xx
0
/CCa
n
.xx
0
/
n
;
(2.5.4)
wherea
0
,...,a
n
areconstants.Differentiating(2.5.4)yields
T
.r/
n
.x
0
/DrŠa
r
; 0rn;
so(2.5.3)determinesa
r
uniquelyas
a
r
D
f
.r/
.x
0
/
; 0rn:
Therefore,
T
n
.x/Df.x
0
/C
f
0
.x
0
/
.xx
0
/CC
f
.n/
.x
0
/
.xx
0
/
n
D
Xn
rD0
f
.r/
.x
0
/
.xx
0
/
r
:
WecallT
n
thenthTaylorpolynomialoffaboutx
0
.
ThefollowingtheoremdescribeshowT
n
approximatesf nearx
0
.
Theorem2.5.1
Iff
.n/
.x
0
/existsforsomeintegern   1andT
n
isthenthTaylor
polynomialoff aboutx
0
;then
lim
x!x
0
f.x/T
n
.x/
.xx
0
/n
D0:
(2.5.5)
100 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Proof
Theproofisbyinduction.LetP
n
betheassertionofthetheorem.From(2.5.2)
weknowthat(2.5.5)istrueifnD1;thatis,P
1
istrue. NowsupposethatP
n
istruefor
someintegern1,andf
.nC1/
exists.Sincetheratio
f.x/T
nC1
.x/
.xx
0
/nC1
isindeterminateoftheform0=0asx!x
0
,L’Hospital’sruleimpliesthat
lim
x!x
0
f.x/T
nC1
.x/
.xx
0
/nC1
D
1
nC1
lim
x!x
0
f0.x/T0
nC1
.x/
.xx
0
/n
(2.5.6)
ifthelimitontherightexists.Butf
0
hasannthderivativeatx
0
,and
T
0
nC1
.x/D
Xn
rD0
f
.rC1/
.x
0
/
.xx
0
/
r
isthenthTaylorpolynomialoffaboutx
0
.Therefore,theinductionassumption,applied
tof0,impliesthat
lim
x!x
0
f0.x/T0
nC1
.x/
.xx
0
/n
D0:
Thisand(2.5.6)implythat
lim
x!x
0
f.x/T
nC1
.x/
.xx
0
/nC1
D0;
whichcompletestheinduction.
Itcanbeshown(Exercise2.5.8)thatif
p
n
Da
0
Ca
1
.xx
0
/CCa
n
.xx
0
/
n
isapolynomialofdegreensuchthat
lim
x!x
0
f.x/p
n
.x/
.xx
0
/n
D0;
then
a
r
D
f
.r/
.x
0
/
I
thatis,p
n
DT
n
.Thus,T
n
istheonlypolynomialofdegreenthatapproximatesf near
x
0
inthemannerindicatedin(2.5.5).
Theorem2.5.1canberestatedasageneralizationofLemma2.3.2.
Lemma2.5.2
Iff
.n/
.x
0
/exists;then
f.x/D
Xn
rD0
f
.r/
.x
0
/
.xx
0
/
r
CE
n
.x/.xx
0
/
n
;
(2.5.7)
where
lim
x!x
0
E
n
.x/DE
n
.x
0
/D0:
Section2.5
Taylor’sTheorem
101
Proof
Define
E
n
.x/D
8
<
:
f.x/T
n
.x/
.xx
0
/n
; x2D
f
fx
0
g;
0;
xDx
0
:
Then(2.5.5)impliesthatlim
x!x
0
E
n
.x/DE
n
.x
0
/D0,anditisstraightforwardtoverify
(2.5.7).
Example2.5.1
Iff.x/Dex,thenf.n/.x/Dex.Therefore,f.n/.0/D1forn0,
sothenthTaylorpolynomialoff aboutx
0
D0is
T
n
.x/D
Xn
rD0
x
r
D1C
x
C
x
2
CC
x
n
:
(2.5.8)
Theorem2.5.1impliesthat
lim
x!0
e
x
Xn
rD0
x
r
xn
D0:
(SeealsoExercise2.4.38.)
Example2.5.2
Iff.x/Dlogx,thenf.1/D0and
f
.r/
.x/D.1/
.r1/
.r1/Š
xr
; r1;
sothenthTaylorpolynomialoff aboutx
0
D1is
T
n
.x/D
Xn
rD1
.1/
r1
r
.x1/
r
ifn1.(T
0
D0.)Theorem2.5.1impliesthat
lim
x!1
logx
Xn
rD1
.1/
r1
r.x1/
r
.x1/n
D0; n1:
Example2.5.3
Iff.x/D.1Cx/
q
,then
f
0
.x/Dq.1Cx/
q1
f
00
.x/Dq.q1/.1Cx/
q2
:
:
:
f
.n/
.x/Dq.q1/.qnC1/.1Cx/
qn
:
102 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Ifwedefine
q
0
!
D1 and
q
n
!
D
q.q1/.qnC1/
; n1;
then
f
.n/
.0/
D
q
n
!
;
andthenthTaylorpolynomialoff about0canbewrittenas
T
n
.x/D
Xn
rD0
q
r
!
x
r
:
(2.5.9)
Theorem2.5.1impliesthat
lim
x!0
.1Cx/
q
Xn
rD0
q
r
!
x
r
xn
D0; n0:
Ifqis anonnegative integer, then
q
n
!
is thebinomialcoefficient definedinExer-
cise1.2.19.Inthiscase,weseefrom(2.5.9)that
T
n
.x/D.1Cx/
q
Df.x/; nq:
Applications toFindingLocal Extrema
Lemma2.5.2yieldsthefollowingtheorem.
Theorem2.5.3
Supposethatf hasnderivativesatx
0
andnisthesmallestpositive
integersuchthatf
.n/
.x
0
/¤0:
(a)
Ifnisodd;x
0
isnotalocalextremepointoff:
(b)
Ifniseven;x
0
isalocalmaximumoff iff
.n/
.x
0
/<0;oralocalmininumoff if
f
.n/
.x
0
/>0:
Proof
Sincef
.r/
.x
0
/D0for1rn1,(2.5.7)impliesthat
f.x/f.x
0
/D
"
f
.n/
.x
0
/
CE
n
.x/
#
.xx
0
/
n
(2.5.10)
insomeintervalcontainingx
0
. Sincelim
x!x
0
E
n
.x/D 0andf
.n/
.x
0
/¤0,thereisa
ı>0suchthat
jE
n
.x/j<
ˇ
ˇ
ˇ
ˇ
ˇ
f
.n/
.x
0
/
ˇ
ˇ
ˇ
ˇ
ˇ
if jxx
0
j<ı:
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