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Section2.1
FunctionsandLimits
43
Throughoutthisbook,“lim
x!x
0
f.x/exists”willmeanthat
lim
x!x
0
f.x/DL; whereLisfinite.
ToleaveopenthepossibilitythatLD˙1,wewillsaythat
lim
x!x
0
f.x/ existsintheextendedreals.
Thisconventionalsoappliestoone-sidedlimitsandlimitsasxapproaches˙1.
WementionedearlierthatTheorems2.1.3and2.1.4remainvalidif“lim
x!x
0
”isre-
placedby“lim
x!x
0
”or“lim
x!x
0
C
.” They y arealsovalidwithx
0
replaced by˙1.
Moreover,thecounterpartsof(2.1.10),(2.1.11),and(2.1.12)inalltheseversionsofThe-
orem2.1.4remainvalidifeitherorbothofL
1
andL
2
areinfinite,providedthattheir
rightsidesarenotindeterminate(Exercises2.1.28and2.1.29). Equation(2.1.14)andits
counterpartsremainvalidifL
1
=L
2
isnotindeterminateandL
2
¤0(Exercise2.1.30).
Example2.1.13
ResultslikeTheorem2.1.4yield
lim
x!1
sinhxD lim
x!1
eex
2
D
1
2
lim
x!1
e
x
 lim
x!1
e
x
D
1
2
.10/D1;
lim
x!1
sinhxD lim
x!1
e
x
e
x
2
D
1
2
lim
x!1
e
x
 lim
x!1
e
x
D
1
2
.01/D1;
and
lim
x!1
e
x
x
D
lim
x!1
e
x
lim
x!1
x
D
0
1
D0:
Example2.1.14
If
f.x/De
2x
e
x
;
wecannotobtainlim
x!1
f.x/bywriting
lim
x!1
f.x/D lim
x!1
e
2x
 lim
x!1
e
x
;
becausethisproducestheindeterminateform11.However,bywriting
f.x/De
2x
.1e
x
/;
wefindthat
lim
x!1
f.x/D
lim
x!1
e
2x

lim
x!1
1 lim
x!1
e
x
D1.10/D1:
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44 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Example2.1.15
Let
g.x/D
2x
2
xC1
3x2C2x1
:
Tryingtofindlim
x!1
g.x/byapplyingaversionofTheorem2.1.4tothisfractionasitis
writtenleadstoanindeterminateform(tryit!).However,byrewritingitas
g.x/D
21=xC1=x
2
3C2=x1=x2
; x¤0;
wefindthat
lim
x!1
g.x/D
lim
x!1
2 lim
x!1
1=xC lim
x!1
1=x
2
lim
x!1
3C lim
x!1
2=x lim
x!1
1=x
2
D
20C0
3C00
D
2
3
:
MonotonicFunction
AfunctionfisnondecreasingonanintervalIif
f.x
1
/f.x
2
/ wheneverx
1
andx
2
areinIandx
1
<x
2
;
(2.1.19)
ornonincreasingonIif
f.x
1
/f.x
2
/ wheneverx
1
andx
2
areinIandx
1
<x
2
:
(2.1.20)
Ineithercase,f isonI.Ifcanbereplacedby<in(2.1.19),f isincreasingonI.If
canbereplacedby>in(2.1.20),f isdecreasingonI.Ineitherofthesetwocases,f f is
strictlymonotoniconI.
Example2.1.16
Thefunction
f.x/D
(
x; 0x<1;
2; 1x2;
isnondecreasingonI DŒ0;2(Figure2.1.4),andf isnonincreasingonI I DŒ0;2.
2
2
1
1
y
x
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Section2.1
FunctionsandLimits
45
Figure2.1.4
Thefunctiong.x/Dx2isincreasingonŒ0;1/(Figure2.1.5),
y
x
y = x2
Figure2.1.5
andh.x/Dx3isdecreasingon.1;1/(Figure2.1.6).
y = − x3
y
x
Figure2.1.6
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46 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Intheproofofthefollowingtheorem,weassumethatyouhaveformulatedthedefinitions
calledforinExercise2.1.19.
Theorem2.1.9
Supposethatf ismonotonicon.a;b/anddefine
˛D
inf
a<x<b
f.x/ and ˇD sup
a<x<b
f.x/:
(a)
Iff isnondecreasing;thenf.aC/D˛andf.b/Dˇ:
(b)
Iff isnonincreasing;thenf.aC/Dˇandf.b/D˛:
.HereaCD1ifaD1andbD1ifbD1:/
(c)
Ifa<x
0
<b,thenf.x
0
C/andf.x
0
/existandarefiniteImoreover;
f.x
0
/f.x
0
/f.x
0
C/
iff isnondecreasing;and
f.x
0
/f.x
0
/f.x
0
C/
iff isnonincreasing:
Proof (a)
Wefirstshowthatf.aC/D˛.If
M > > ˛, , thereisanx
0
in.a;b/suchthatf.x
0
/ < M. Sincef isnondecreasing,
f.x/<Mifa<x <x
0
. Therefore,if˛D1,thenf.aC/D1. If˛>1,let
MD˛C,where>0.Then˛f.x/<˛C,so
jf.x/˛j< if a<x<x
0
:
(2.1.21)
IfaD1,thisimpliesthatf.1/D˛. Ifa>1,letıDx
0
a. Then(2.1.21)is
equivalentto
jf.x/˛j< if a<x<aCı;
whichimpliesthatf.aC/D˛.
Wenowshowthatf.b/Dˇ.IfM<ˇ,thereisanx
0
in.a;b/suchthatf.x
0
/>M.
Sincef is s nondecreasing, f.x/ > M ifx
0
< x < < b. Therefore, , ifˇ D 1, , then
f.b/D1.Ifˇ<1,letMDˇ,where>0.Thenˇ<f.x/ˇ,so
jf.x/ˇj< if x
0
<x<b:
(2.1.22)
Ifb D D 1,thisimpliesthatf.1/ D ˇ. Ifb b < 1,letı D bx
0
. Then(2.1.22)is
equivalentto
jf.x/ˇj< if bı<x<b;
whichimpliesthatf.b/Dˇ.
(b)
Theproofissimilartotheproofof
(a)
(Exercise2.1.34).
(c)
Supposethatf isnondecreasing. . Applying
(a)
tof on.a;x
0
/and.x
0
;b/sepa-
ratelyshowsthat
f.x
0
/D
sup
a<x<x
0
f.x/ and f.x
0
C/D
inf
x
0
<x<b
f.x/:
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Section2.1
FunctionsandLimits
47
However,ifx
1
<x
0
<x
2
,then
f.x
1
/f.x
0
/f.x
2
/I
hence,
f.x
0
/f.x
0
/f.x
0
C/:
Weleavethecasewheref isnonincreasingtoyou(Exercise2.1.34).
LimitsInferiorandSuperior
Wenowintroducesomeconceptsrelatedtolimits. Weleavethestudyoftheseconcepts
mainlytotheexercises.
Wesaythatf isboundedonasetSifthereisaconstantM<1suchthatjf.x/jM
forallxinS.
Definition2.1.10
SupposethatfisboundedonŒa;x
0
/,wherex
0
maybefiniteor1.
Forax<x
0
,define
S
f
.xIx
0
/D
sup
xt<x
0
f.t/
and
I
f
.xIx
0
/D
inf
xt<x
0
f.t/:
Thentheleftlimitsuperioroff atx
0
isdefinedtobe
lim
x!x
0
f.x/D
lim
x!x
0
S
f
.xIx
0
/;
andtheleftlimitinferioroff atx
0
isdefinedtobe
lim
x!x
0
f.x/D
lim
x!x
0
I
f
.xIx
0
/:
(Ifx
0
D1,wedefinex
0
D1.)
Theorem2.1.11
Iff isboundedonŒa;x
0
/;thenˇ D
lim
x!x
0
f.x/existsandis
theuniquerealnumberwiththefollowingpropertiesW
(a)
If>0,thereisana
1
inŒa;x
0
/suchthat
f.x/<ˇC if a
1
x<x
0
:
(2.1.23)
(b)
If>0anda
1
isinŒa;x
0
/;then
f.
x/>ˇ forsome
x2Œa
1
;x
0
/:
Proof
Sincef is s boundedonŒa;x
0
/, S
f
.xIx
0
/is nonincreasingandboundedon
Œa;x
0
/. ByapplyingTheorem2.1.9
(b)
toS
f
.xIx
0
/,weconcludethatˇexists(finite).
Therefore,if>0,thereisan
ainŒa;x
0
/suchthat
ˇ=2<S
f
.xIx
0
/<ˇC=2 if
ax<x
0
:
(2.1.24)
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48 Chapter2
DifferentialCalculusofFunctionsofOneVariable
SinceS
f
.xIx
0
/isanupperboundof
˚
f.t/
ˇ
ˇ
xt<x
0
,f.x/S
f
.xIx
0
/.Therefore,
thesecondinequalityin(2.1.24)implies(2.1.23)witha
1
D
a.Thisproves
(a)
.Toprove
(b)
, leta
1
begivenanddefinex
1
D max.a
1
;
a/. Thenthefirstinequalityin(2.1.24)
impliesthat
S
f
.x
1
Ix
0
/>ˇ=2:
(2.1.25)
SinceS
f
.x
1
Ix
0
/isthesupremumof
˚
f.t/
ˇ
ˇ
x
1
<t<x
0
,thereisan
xinŒx
1
;x
0
/such
that
f.
x/>S
f
.x
1
Ix
0
/=2:
Thisand(2.1.25)implythatf.
x/>ˇ.Since
xisinŒa
1
;x
0
/,thisproves
(b)
.
Nowweshowthattherecannotbemorethanonerealnumberwithproperties
(a)
and
(b)
. Supposethatˇ
1
2
andˇ
2
hasproperty
(b)
;thus,if>0anda
1
isinŒa;x
0
/,
thereisan
xinŒa
1
;x
0
/suchthatf.
x/>ˇ
2
.LettingDˇ
2
ˇ
1
,weseethatthere
isan
xinŒa
1
;b/suchthat
f.
x/>ˇ
2
.ˇ
2
ˇ
1
/Dˇ
1
;
soˇ
1
cannothaveproperty
(a)
. Therefore,therecannotbemorethanonerealnumber
thatsatisfiesboth
(a)
and
(b)
.
Theproofofthefollowingtheoremissimilartothis(Exercise2.1.35).
Theorem2.1.12
Iff isboundedonŒa;x
0
/;then˛Dlim
x!x
0
f.x/existsandis
theuniquerealnumberwiththefollowingproperties:
(a)
If>0;thereisana
1
inŒa;x
0
/suchthat
f.x/>˛ if a
1
x<x
0
:
(b)
If>0anda
1
isinŒa;x
0
/;then
f.
x/<˛C forsome
x2Œa
1
;x
0
/:
2.1Exercises
1.
Eachofthefollowingconditionsfailstodefineafunctiononanydomain. State
why.
(a)
sinf.x/Dx
(b)
e
f.x/
Djxj
(c)
1Cx
2
CŒf.x/
2
D0
(d)
f.x/Œf.x/1Dx
2
2.
If
f.x/D
r
.x3/.xC2/
x1
and
g.x/D
x
2
16
x7
p
x29;
findD
f
,D
f˙g
,D
fg
,andD
f=g
.
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Section2.1
FunctionsandLimits
49
3.
FindD
f
.
(a)
f.x/Dtanx
(b)
f.x/D
1
p
1jsinxj
(c)
f.x/D
1
x.x1/
(d)
f.x/D
sinx
x
(e)
e
Œf.x/
2
Dx; f.x/0
4.
Findlim
x!x
0
f.x/,andjustifyyouranswerswithan–ıproof.
(a)
x2C2xC1; x
0
D1
(b)
x
3
8
x2
; x
0
D2
(c)
1
x21
; x
0
D0
(d)
p
x; x
0
D4
(e)
x
3
1
.x1/.x2/
Cx; x
0
D1
5.
ProvethatDefinition2.1.2isunchangedifEqn.(2.1.4)isreplacedby
jf.x/Lj<K;
whereKisanypositiveconstant.(Thatis,lim
x!x
0
f.x/DLaccordingtoDefini-
tion2.1.2ifandonlyiflim
x!x
0
f.x/DLaccordingtothemodifieddefinition.)
6.
UseTheorem2.1.4andtheknownlimitslim
x!x
0
xDx
0
,lim
x!x
0
c Dctofind
theindicatedlimits.
(a)
lim
x!2
x
2
C2xC3
2x3C1
(b)
lim
x!2
1
xC1
1
x1
(c)
lim
x!1
x1
x3Cx22x
(d)
lim
x!1
x
8
1
x41
7.
Findlim
x!x
0
f.x/andlim
x!x
0
C
f.x/,iftheyexist. Use–ıproofs,whereap-
plicable,tojustifyyouranswers.
(a)
xCjxj
x
; x
0
D0
(b)
xcos
1
x
Csin
1
x
Csin
1
jxj
; x
0
D0
(c)
jx1j
x2Cx2
; x
0
D1
(d)
x
2
Cx2
p
xC2
; x
0
D2
8.
Prove:Ifh.x/0fora<x<x
0
andlim
x!x
0
h.x/exists,thenlim
x!x
0
h.x/
 0.Concludefromthisthatiff
2
.x/f
1
.x/fora<x<x
0
,then
lim
x!x
0
f
2
.x/ lim
x!x
0
f
1
.x/
ifbothlimitsexist.
50 Chapter2
DifferentialCalculusofFunctionsofOneVariable
9. (a)
Prove:Iflim
x!x
0
f.x/exists,thereisaconstantManda>0suchthat
jf.x/j  M if0 < jxx
0
j < . (Wesaythenthatf isboundedon
˚
x
ˇ
ˇ
0<jxx
0
j<
.)
(b)
Statesimilarresultswith“lim
x!x
0
”replacedby“lim
x!x
0
.”
(c)
Statesimilarresultswith“lim
x!x
0
”replacedby“lim
x!x
0
C
.”
10.
Supposethatlim
x!x
0
f.x/DLandnisapositiveinteger.Provethatlim
x!x
0
Œf.x/
n
D
L
n
(a)
byusingTheorem2.1.4andinduction;
(b)
directlyfromDefinition2.1.2.
H
INT
:YouwillfindExercise2.1.9usefulfor.b/:
11.
Prove:Iflim
x!x
0
f.x/DL>0,thenlim
x!x
0
p
f.x/D
p
L.
12.
ProveTheorem2.1.6.
13. (a)
UsingthehintstatedafterTheorem2.1.6,provethatTheorem2.1.3remains
validwith“lim
x!x
0
”replacedby“lim
x!x
0
.”
(b)
Repeat
(a)
forTheorem2.1.4.
14.
Definethestatement“lim
x!1
f.x/DL.”
15.
Findlim
x!1
f.x/ifitexists,andjustifyyouranswerdirectlyfromDefinition2.1.7.
(a)
1
x2C1
(b)
sinx
jxj˛
.˛>0/
(c)
sinx
jxj˛
.˛0/
(d)
e
x
sinx
(e)
tanx
(f)
e
x
2
e
2x
16.
Theorems 2.1.3and2.1.4remain n validwith“lim
x!x
0
”replaced throughoutby
“lim
x!1
”(“lim
x!1
”).Howwouldtheirproofshavetobechanged?
17.
UsingthedefinitionyougaveinExercise2.1.14,showthat
(a)
lim
x!1
1
1
x2
D1
(b)
lim
x!1
2jxj
1Cx
D2
(c)
lim
x!1
sinxdoesnotexist
18.
Findlim
x!1
f.x/,ifitexists,foreachfunctioninExercise2.1.15.Justifyyour
answersdirectlyfromthedefinitionyougaveinExercise2.1.14.
19.
Define
(a)
lim
x!x
0
f.x/D1
(b)
lim
x!x
0
C
f.x/D1
(c)
lim
x!x
0
C
f.x/D1
20.
Find
(a)
lim
x!0C
1
x3
(b)
lim
x!0
1
x3
(c)
lim
x!0C
1
x6
(d)
lim
x!0
1
x6
(e)
lim
x!x
0
C
1
.xx
0
/2k
(f)
lim
x!x
0
1
.xx
0
/2kC1
(kDpositiveinteger)
Section2.1
FunctionsandLimits
51
21.
Define
(a)
lim
x!x
0
f.x/D1
(b)
lim
x!x
0
f.x/D1
22.
Find
(a)
lim
x!0
1
x3
(b)
lim
x!0
1
x6
(c)
lim
x!x
0
1
.xx
0
/2k
(d)
lim
x!x
0
1
.xx
0
/2kC1
(kDpositiveinteger)
23.
Define
(a)
lim
x!1
f.x/D1
(b)
lim
x!1
f.x/D1
24.
Find
(a)
lim
x!1
x
2k
(b)
lim
x!1
x
2k
(c)
lim
x!1
x
2kC1
(d)
lim
x!1
x
2kC1
(k=positiveinteger)
(e)
lim
x!1
p
xsinx
(f)
lim
x!1
e
x
25.
Supposethat f and d garedefined on.a;1/ and.c;1/ respectively, , andthat
g.x/>aifx>c. Supposealsothatlim
x!1
f.x/DL,where1L1,
andlim
x!1
g.x/D1.Showthatlim
x!1
f.g.x//DL.
26. (a)
Prove:lim
x!x
0
f.x/doesnotexist(finite)ifforsome
0
>0,everydeleted
neighborhoodofx
0
containspointsx
1
andx
2
suchthat
jf.x
1
/f.x
2
/j
0
:
(b)
Giveanalogousconditionsforthenonexistenceof
lim
x!x
0
C
f.x/;
lim
x!x
0
f.x/;
lim
x!1
f.x/; and
lim
x!1
f.x/:
27.
Prove: If1< x
0
<1,thenlim
x!x
0
f.x/existsintheextendedrealsifand
onlyiflim
x!x
0
f.x/andlim
x!x
0
C
f.x/bothexistintheextendedrealsandare
equal,inwhichcaseallthreeareequal.
InExercises2.1.282.1.30consideronlythecasewhereatleastoneofL
1
andL
2
is˙1.
28.
Prove:Iflim
x!x
0
f.x/DL
1
,lim
x!x
0
g.x/DL
2
,andL
1
CL
2
isnotindetermi-
nate,then
lim
x!x
0
.fCg/.x/DL
1
CL
2
:
52 Chapter2
DifferentialCalculusofFunctionsofOneVariable
29.
Prove:Iflim
x!1
f.x/DL
1
,lim
x!1
g.x/DL
2
,andL
1
L
2
isnotindeterminate,
then
lim
x!1
.fg/.x/DL
1
L
2
:
30. (a)
Prove:Iflim
x!x
0
f.x/DL
1
,lim
x!x
0
g.x/DL
2
¤0,andL
1
=L
2
isnot
indeterminate,then
lim
x!x
0
f
g
.x/D
L
1
L
2
:
(b)
ShowthatitisnecessarytoassumethatL
2
¤0in
(a)
byconsideringf.x/D
sinx,g.x/Dcosx,andx
0
D=2.
31.
Find
(a)
lim
x!0C
x
3
C2xC3
2x4C3x2C2
(b)
lim
x!0
x
3
C2xC3
2x4C3x2C2
(c)
lim
x!1
2x
4
C3x
2
C2
x3C2xC3
(d)
lim
x!1
2x
4
C3x
2
C2
x3C2xC3
(e)
lim
x!1
.e
x
2
e
x
/
(f)
lim
x!1
xC
p
xsinx
2xCex
32.
Findlim
x!1
r.x/andlim
x!1
r.x/fortherationalfunction
r.x/D
a
0
Ca
1
xCCa
n
x
n
b
0
Cb
1
xCCb
m
xm
;
wherea
n
¤0andb
m
¤0.
33.
Supposethatlim
x!x
0
f.x/existsforeveryx
0
in.a;b/andg.x/Df.x/except
onasetSwithnolimitpointsin.a;b/.Whatcanbesaidaboutlim
x!x
0
g.x/for
x
0
in.a;b/?Justifyyouranswer.
34.
ProveTheorem2.1.9
(b)
,andcompletetheproofofTheorem2.1.9
(b)
inthecase
wheref isnonincreasing.
35.
ProveTheorem2.1.12.
36.
Showthatiff isboundedonŒa;x
0
/,then
(a)
lim
x!x
0
f.x/
lim
x!x
0
f.x/.
(b)
lim
x!x
0
.f/.x/D
lim
x!x
0
f.x/and
lim
x!x
0
.f/.x/D lim
x!x
0
f.x/.
(c)
lim
x!x
0
f.x/ D
lim
x!x
0
f.x/ifandonlyiflim
x!x
0
f.x/exists,inwhich
case
lim
x!x
0
f.x/D
lim
x!x
0
f.x/D
lim
x!x
0
f.x/:
37.
Supposethatf andgareboundedonŒa;x
0
/.
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