Lesson 3: The Concept of Limit

. . . . . .

Section1.3TheConceptofLimit

V63.0121.006/016, CalculusI

January26, 2009

Announcements

I BlackboardsitesareupI OfficeHours: MW 1:30–2:30, R 9–10(CIWW 726)I WebAssignmentsnotdueuntilFeb2(butthereareseveral)

Limit

. . . . . .

. . . . . .

Zeno’sParadox

Thatwhichisinlocomotionmustarriveatthehalf-waystagebeforeitarrivesatthegoal.

(Aristotle Physics VI:9,239b10)

. . . . . .

Outline

Heuristics

Errorsandtolerances

Examples

Pathologies

PreciseDefinitionofaLimit

. . . . . .

HeuristicDefinitionofaLimit

DefinitionWewrite

limx→a

f(x) = L

andsay

“thelimitof f(x), as x approaches a, equals L”

ifwecanmakethevaluesof f(x) arbitrarilycloseto L (asclosetoL aswelike)bytaking x tobesufficientlycloseto a (oneithersideof a)butnotequalto a.

. . . . . .

Outline

Heuristics

Errorsandtolerances

Examples

Pathologies


. . . . . .

Theerror-tolerancegame

A gamebetweentwoplayerstodecideifalimit limx→a

f(x) exists.

Step1 Player1: Choose L tobethelimit.

Step2 Player2: Proposean“error”levelaround L.

Step3 Player1: Choosea“tolerance”levelaround a sothatx-pointswithinthattolerancelevelof a aretakentoy-valueswithintheerrorlevelof L, withthepossibleexceptionof a itself.

Step4 GobacktoStep2untilPlayer1cannotmove.

IfPlayer1canalwaysfindatolerancelevel, limx→a

f(x) = L.

. . . . . .


.

.Thistoleranceistoobig.Stilltoobig.Thislooksgood.Sodoesthis

.a

.L

I Tobelegit, thepartofthegraphinsidetheblue(vertical)stripmustalsobeinsidethegreen(horizontal)strip.

I IfPlayer2shrinkstheerror, Player1canstillwin.

. . . . . .


.

.Thistoleranceistoobig

.Stilltoobig.Thislooksgood.Sodoesthis

.a

.L



. . . . . .


.


.a

.L



. . . . . .


.

.Thistoleranceistoobig

.Stilltoobig

.Thislooksgood.Sodoesthis

.a

.L



. . . . . .


.


.a

.L



. . . . . .


.

.Thistoleranceistoobig.Stilltoobig

.Thislooksgood

.Sodoesthis

.a

.L



. . . . . .


.

.Thistoleranceistoobig.Stilltoobig.Thislooksgood

.Sodoesthis

.a

.L



. . . . . .


.


.a

.L



. . . . . .

Outline

Heuristics

Errorsandtolerances

Examples

Pathologies


. . . . . .

ExampleFind lim

x→0x2 ifitexists.

Solution

I I claimthelimitiszero.I Iftheerrorlevelis 0.01, I needtoguaranteethat

−0.01 < x2 < 0.01 forall x sufficientlyclosetozero.I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, soI winthatround.I Whatshouldthetolerancebeiftheerroris 0.0001?

Bysettingtoleranceequaltothesquarerootoftheerror, wecanguaranteetobewithinanyerror.

. . . . . .

ExampleFind lim

x→0x2 ifitexists.

Solution

I I claimthelimitiszero.

I Iftheerrorlevelis 0.01, I needtoguaranteethat−0.01 < x2 < 0.01 forall x sufficientlyclosetozero.

I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, soI winthatround.I Whatshouldthetolerancebeiftheerroris 0.0001?


. . . . . .

ExampleFind lim

x→0x2 ifitexists.

Solution


−0.01 < x2 < 0.01 forall x sufficientlyclosetozero.

I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, soI winthatround.I Whatshouldthetolerancebeiftheerroris 0.0001?


. . . . . .

ExampleFind lim

x→0x2 ifitexists.

Solution


−0.01 < x2 < 0.01 forall x sufficientlyclosetozero.I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, soI winthatround.

I Whatshouldthetolerancebeiftheerroris 0.0001?


. . . . . .

ExampleFind lim

x→0x2 ifitexists.

Solution


−0.01 < x2 < 0.01 forall x sufficientlyclosetozero.I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, soI winthatround.I Whatshouldthetolerancebeiftheerroris 0.0001?


. . . . . .

Example

Find limx→0

|x|x

ifitexists.

Solution

Thefunctioncanalsobewrittenas

|x|x

=

{1 if x > 0;

−1 if x < 0

Whatwouldbethelimit?Theerror-tolerancegamefails, but

limx→0+

f(x) = 1 limx→0−

f(x) = −1

. . . . . .

Example

Find limx→0

|x|x

ifitexists.

SolutionThefunctioncanalsobewrittenas

|x|x

=

{1 if x > 0;

−1 if x < 0

Whatwouldbethelimit?

Theerror-tolerancegamefails, but

limx→0+

f(x) = 1 limx→0−

f(x) = −1

. . . . . .


. .x

.y

..−1

..1 .

.

.Part of graph in-side blue is notinside green

.Part of graph in-side blue is notinside green

I Thesearetheonlygoodchoices; thelimitdoesnotexist.

. . . . . .

One-sidedlimits

DefinitionWewrite

limx→a+

f(x) = L

andsay

“thelimitof f(x), as x approaches a fromthe right, equals L”

ifwecanmakethevaluesof f(x) arbitrarilycloseto L (asclosetoL aswelike)bytaking x tobesufficientlycloseto a and greaterthan a.

. . . . . .

One-sidedlimits

DefinitionWewrite

limx→a−

f(x) = L

andsay

“thelimitof f(x), as x approaches a fromthe left, equals L”

ifwecanmakethevaluesof f(x) arbitrarilycloseto L (asclosetoL aswelike)bytaking x tobesufficientlycloseto a and less thana.

. . . . . .


. .x

.y

..−1

..1 .

.

.Part of graph in-side blue is in-side green

.Part of graph in-side blue is in-side green

I So limx→0+

f(x) = 1 and limx→0−

f(x) = −1

. . . . . .

Example

Find limx→0

|x|x

ifitexists.

SolutionThefunctioncanalsobewrittenas

|x|x

=

{1 if x > 0;

−1 if x < 0

Whatwouldbethelimit?Theerror-tolerancegamefails, but

limx→0+

f(x) = 1 limx→0−

f(x) = −1

. . . . . .

Example

Find limx→0+

1xifitexists.

SolutionThelimitdoesnotexistbecausethefunctionisunboundednear0. Nextweekwewillunderstandthestatementthat

limx→0+

1x= +∞

. . . . . .


. .x

.y

.0

..L?

.The graph escapes thegreen, so no good.Evenworse!

.The limit does not existbecause the function isunbounded near 0

. . . . . .


. .x

.y

.0

..L?

.The graph escapes thegreen, so no good

.Evenworse!


. . . . . .


. .x

.y

.0

..L?



. . . . . .


. .x

.y

.0

..L?

.The graph escapes thegreen, so no good

.Evenworse!


. . . . . .


. .x

.y

.0

..L?



. . . . . .

Example

Find limx→0+

1xifitexists.

SolutionThelimitdoesnotexistbecausethefunctionisunboundednear0. Nextweekwewillunderstandthestatementthat

limx→0+

1x= +∞

. . . . . .

Weird, wildstuff

ExampleFind lim

x→0sin

(πx

)ifitexists.

I f(x) = 0 when x =

1kforanyinteger k

I f(x) = 1 when x =

24k+ 1

foranyinteger k

I f(x) = −1 when x =

24k− 1

foranyinteger k

. . . . . .

Functionvalues

x π/x sin(π/x)1 π 01/2 2π 01/k kπ 02 π/2 12/5 5π/2 12/9 9π/2 12/13 13π/2 12/3 3π/2 −12/7 7π/2 −12/11 11π/2 −1

.

..π/2

..π

..3π/2

. .0

. . . . . .

Weird, wildstuff

ExampleFind lim

x→0sin

(πx

)ifitexists.

I f(x) = 0 when x =

1kforanyinteger k

I f(x) = 1 when x =

24k+ 1

foranyinteger k


24k− 1

foranyinteger k

. . . . . .

Weird, wildstuff

ExampleFind lim

x→0sin

(πx

)ifitexists.

I f(x) = 0 when x =1kforanyinteger k

I f(x) = 1 when x =

24k+ 1

foranyinteger k


24k− 1

foranyinteger k

. . . . . .

Weird, wildstuff

ExampleFind lim

x→0sin

(πx

)ifitexists.


I f(x) = 1 when x =2

4k+ 1foranyinteger k


24k− 1

foranyinteger k

. . . . . .

Weird, wildstuff

ExampleFind lim

x→0sin

(πx

)ifitexists.


I f(x) = 1 when x =2

4k+ 1foranyinteger k

I f(x) = −1 when x =2

4k− 1foranyinteger k

. . . . . .

Weird, wildstuffcontinued

Hereisagraphofthefunction:

. .x

.y

..−1

..1

Thereareinfinitelymanypointsarbitrarilyclosetozerowheref(x) is 0, or 1, or −1. Sothelimitcannotexist.

. . . . . .

Outline

Heuristics

Errorsandtolerances

Examples

Pathologies


. . . . . .

Whatcouldgowrong?SummaryofLimitPathologies

Howcouldafunctionfailtohavealimit? Somepossibilities:I left-andright-handlimitsexistbutarenotequalI Thefunctionisunboundednear aI Oscillationwithincreasinglyhighfrequencynear a

. . . . . .

MeettheMathematician: AugustinLouisCauchy

I French, 1789–1857I RoyalistandCatholicI madecontributionsingeometry, calculus,complexanalysis,numbertheory

I createdthedefinitionoflimitweusetodaybutdidn’tunderstandit

. . . . . .

Outline

Heuristics

Errorsandtolerances

Examples

Pathologies


. . . . . .

PreciseDefinitionofaLimitNo, thisisnotgoingtobeonthetest

Let f beafunctiondefinedonansomeopenintervalthatcontainsthenumber a, exceptpossiblyat a itself. Thenwesaythatthe limitof f(x) as x approaches a is L, andwewrite

limx→a

f(x) = L,

ifforevery ε > 0 thereisacorresponding δ > 0 suchthat

if 0 < |x− a| < δ, then |f(x)− L| < ε.

. . . . . .

Theerror-tolerancegame= ε, δ

.

.L+ ε

.L− ε

.a− δ .a+ δ

.This δ istoobig

.a− δ.a+ δ

.This δ looksgood

.a− δ.a+ δ

.Sodoesthis δ

.a

.L

Lesson 3: The Concept of Limit

Education

Transcript of Lesson 3: The Concept of Limit