lessons 1–3: functions and the concept of limit

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Section1.1–1.3Functionsandtheconceptoflimit

V63.0121, CalculusI

September9, 2009

Announcements

I SyllabusisonthecommonBlackboardI OfficeHoursTBAI ReadSections1.1–1.3ofthetextbookthisweek.

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Outline

FunctionsFunctionsexpressedbyformulasFunctionsexpressedbydataFunctionsdescribedgraphicallyFunctionsdescribedverballyClassesofFunctions

LimitsHeuristicsErrorsandtolerancesExamplesPathologies

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Whatisafunction?

DefinitionA function f isarelationwhichassignstotoeveryelement x inaset D asingleelement f(x) inaset E.

I Theset D iscalledthe domain of f.I Theset E iscalledthe target of f.I Theset { f(x) | x ∈ D } iscalledthe range of f.

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TheModelingProcess

...Real-worldProblems

..Mathematical

Model

..MathematicalConclusions

..Real-worldPredictions

.model.solve

.interpret

.test

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Plato’sCave

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Functionsexpressedbyformulas

Anyexpressioninasinglevariable x definesafunction. Inthiscase, thedomainisunderstoodtobethelargestsetof x whichaftersubstitution, givearealnumber.

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Functionsexpressedbydata

Inscience, functionsareoftendefinedbydata. Or, weobservedataandassumethatit’sclosetosomenicecontinuousfunction.

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Example

HereisthetemperatureinBoise, Idahomeasuredin15-minuteintervalsovertheperiodAugust22–29, 2008.

...8/22

..8/23

..8/24

..8/25

..8/26

..8/27

..8/28

..8/29

..10

..20

..30

..40

..50

..60

..70

..80

..90

..100

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Functionsdescribedgraphically

Sometimesallwehaveisthe“picture”ofafunction, bywhichwemean, itsgraph.

.

.

Theoneontherightisarelationbutnotafunction.

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Functionsdescribedverbally

Oftentimesourfunctionscomeoutofnatureandhaveverbaldescriptions:

I Thetemperature T(t) inthisroomattime t.I Theelevation h(θ) ofthepointontheequationatlongitude

θ.I Theutility u(x) I derivebyconsuming x burritos.

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ClassesofFunctions

I linearfunctions, definedbyslopeanintercept, pointandpoint, orpointandslope.

I quadraticfunctions, cubicfunctions, powerfunctions,polynomials

I rationalfunctionsI trigonometricfunctionsI exponential/logarithmicfunctions

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Outline

FunctionsFunctionsexpressedbyformulasFunctionsexpressedbydataFunctionsdescribedgraphicallyFunctionsdescribedverballyClassesofFunctions

LimitsHeuristicsErrorsandtolerancesExamplesPathologies

Limit

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Zeno’sParadox

Thatwhichisinlocomotionmustarriveatthehalf-waystagebeforeitarrivesatthegoal.

(Aristotle Physics VI:9,239b10)

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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.

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Theerror-tolerancegame

A gamebetweentwoplayerstodecideifalimit limx→a

f(x) exists.

I Player1: Choose L tobethelimit.I Player2: Proposean“error”levelaround L.I Player1: Choosea“tolerance”levelaround a sothat

x-pointswithinthattolerancelevelaretakento y-valueswithintheerrorlevel.

IfPlayer1canalwayswin, limx→a

f(x) = L.

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.

.Thistoleranceistoobig.Stilltoobig.Thislooksgood.Sodoesthis

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

I IfPlayer2shrinkstheerror, Player1canstillwin.

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.

.Thistoleranceistoobig

.Stilltoobig.Thislooksgood.Sodoesthis

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.Thistoleranceistoobig

.Stilltoobig

.Thislooksgood.Sodoesthis

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.Thistoleranceistoobig.Stilltoobig

.Thislooksgood

.Sodoesthis

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.Thistoleranceistoobig.Stilltoobig.Thislooksgood

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ExampleFind lim

x→0x2 ifitexists.

SolutionBysettingtoleranceequaltothesquarerootoftheerror, wecanguaranteetobewithinanyerror.

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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

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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

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. .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.

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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 (oneithersideof a)and greater than a.

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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 (oneithersideof a)and less than a.

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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

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Example

Find limx→0+

1xifitexists.

SolutionThelimitdoesnotexistbecausethefunctionisunboundednear0. Nextweekwewillunderstandthestatementthat

limx→0+

1x

= +∞

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. .x

.y

.0

..L?

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

.The limit does not existbecause the function isunbounded near 0

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. .x

.y

.0

..L?

.The graph escapes thegreen, so no good

.Evenworse!


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. .x

.y

.0

..L?



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. .x

.y

.0

..L?

.The graph escapes thegreen, so no good

.Evenworse!


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. .x

.y

.0

..L?



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Example

Find limx→0+

1xifitexists.

SolutionThelimitdoesnotexistbecausethefunctionisunboundednear0. Nextweekwewillunderstandthestatementthat

limx→0+

1x

= +∞

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ExampleFind lim

x→0sin

(π

x

)ifitexists.

. .x

.y

..−1

..1

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Whatcouldgowrong?

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

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MeettheMathematician: AugustinLouisCauchy

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

I createdthedefinitionoflimitweusetodaybutdidn’tunderstandit

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PreciseDefinitionofaLimit

No, thisisnotgoingtobeonthetestLet f beafunctiondefinedonansomeopenintervalthatcontainsthenumber a, exceptpossiblyat a itself. Thenwesaythatthe limitof f(x) as xapproaches 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.

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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

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lessons 1–3: functions and the concept of limit

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