lesson03 the concept of limit 027 slides

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Section1.2–1.3A CatalogofEssentialFunctions

TheLimitofaFunction

V63.0121.027, CalculusI

September10, 2009

Announcements

I SyllabusisonthecommonBlackboardI OfficeHoursMTWR 3–4pmI ReadSections1.1–1.3ofthetextbookthisweek.

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Outline

ClassesofFunctionsLinearfunctionsQuadraticfunctionsCubicfunctionsOtherpowerfunctionsRationalfunctionsTrigonometricFunctionsExponentialandLogarithmicfunctions

TransformationsofFunctions

CompositionsofFunctions

LimitsHeuristicsErrorsandtolerancesExamplesPathologies

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ClassesofFunctions

I linearfunctions, definedbyslopeanintercept, pointandpoint, orpointandslope.

I quadraticfunctions, cubicfunctions, powerfunctions,polynomials

I rationalfunctionsI trigonometricfunctionsI exponential/logarithmicfunctions

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Linearfunctions

Linearfunctionshaveaconstantrateofgrowthandareoftheform

f(x) = mx + b.

ExampleInNewYorkCitytaxiscost$2.50togetinand$0.40per 1/5mile. Writethefare f(x) asafunctionofdistance x traveled.

AnswerIf x isinmilesand f(x) indollars,

f(x) = 2.5 + 2x

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Quadraticfunctions

Thesetaketheform

f(x) = ax2 + bx + c

Thegraphisaparabolawhichopensupwardif a > 0, downwardif a < 0.

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Cubicfunctions

Thesetaketheform

f(x) = ax3 + bx2 + cx + d

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Otherpowerfunctions

I Wholenumberpowers: f(x) = xn.

I negativepowersarereciprocals: x−3 =1x3.

I fractionalpowersareroots: x1/3 = 3√x.

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Rationalfunctions

DefinitionA rationalfunction isaquotientofpolynomials.

Example

Thefunction f(x) =x3(x + 3)

(x + 2)(x− 1)isrational.

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TrigonometricFunctions

I SineandcosineI TangentandcotangentI Secantandcosecant

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ExponentialandLogarithmicfunctions

I exponentialfunctions(forexample f(x) = 2x)I logarithmicfunctionsaretheirinverses(forexample

f(x) = log2(x))

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Outline





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Takethe sinefunction andgraphthesetransformations:

I sin(x +

π

2

)I sin

(x− π

2

)I sin (x) +

π

2I sin (x) − π

2

Observethatifthefiddlingoccurswithinthefunction, atransformationisappliedonthe x-axis. Afterthefunction, tothey-axis.

http://tinyurl.com/m9y4e5

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Takethe sinefunction andgraphthesetransformations:

I sin(x +

π

2

)I sin

(x− π

2

)I sin (x) +

π

2I sin (x) − π

2Observethatifthefiddlingoccurswithinthefunction, atransformationisappliedonthe x-axis. Afterthefunction, tothey-axis.

http://tinyurl.com/m9y4e5

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VerticalandHorizontalShifts

Suppose c > 0. ToobtainthegraphofI y = f(x) + c, shiftthegraphof y = f(x) adistance c units

upward

I y = f(x) − c, shiftthegraphof y = f(x) adistance c units

downward

I y = f(x− c), shiftthegraphof y = f(x) adistance c units

totheright

I y = f(x + c), shiftthegraphof y = f(x) adistance c units

totheleft

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Suppose c > 0. ToobtainthegraphofI y = f(x) + c, shiftthegraphof y = f(x) adistance c unitsupward

I y = f(x) − c, shiftthegraphof y = f(x) adistance c units

downward


totheright


totheleft

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I y = f(x) − c, shiftthegraphof y = f(x) adistance c unitsdownward


totheright


totheleft

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I y = f(x− c), shiftthegraphof y = f(x) adistance c units totheright


totheleft

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I y = f(x− c), shiftthegraphof y = f(x) adistance c units totheright

I y = f(x + c), shiftthegraphof y = f(x) adistance c units totheleft

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Outline





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Compositionisacompoundingoffunctionsinsuccession

..f .g

.g ◦ f

.x .(g ◦ f)(x).f(x)

.

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Composing

ExampleLet f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.

Solutionf ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2). Notetheyare not thesame.

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Decomposing

ExampleExpress

√x2 − 4 asacompositionoftwofunctions. Whatisits

domain?

SolutionWecanwritetheexpressionas f ◦ g, where f(u) =

√u and

g(x) = x2 − 4. Therangeof g needstobewithinthedomainof f.Toinsurethat x2 − 4 ≥ 0, wemusthave x ≤ −2 or x ≥ 2.

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TheFarSide

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Outline





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

.a

.L

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

I IfPlayer2shrinkstheerror, Player1canstillwin.

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.

.Thistoleranceistoobig

.Stilltoobig.Thislooksgood.Sodoesthis

.a

.L



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.L



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.

.Thistoleranceistoobig

.Stilltoobig

.Thislooksgood.Sodoesthis

.a

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.L



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.

.Thistoleranceistoobig.Stilltoobig

.Thislooksgood

.Sodoesthis

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.Thistoleranceistoobig.Stilltoobig.Thislooksgood

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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.

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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?


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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?


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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?


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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?


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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.

. . . . . .

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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Weird, wildstuff

ExampleFind lim

x→0sin

(π

x

)ifitexists.

I f(x) = 0 when x =

1kforanyinteger k

I f(x) = 1 when x =

12k + 1/2

foranyinteger k

I f(x) = −1 when x =

12k− 1/2

foranyinteger k

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Weird, wildstuff

ExampleFind lim

x→0sin

(π

x

)ifitexists.

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

I f(x) = 1 when x =

12k + 1/2

foranyinteger k


12k− 1/2

foranyinteger k

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Weird, wildstuff

ExampleFind lim

x→0sin

(π

x

)ifitexists.


I f(x) = 1 when x =1

2k + 1/2foranyinteger k


12k− 1/2

foranyinteger k

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Weird, wildstuff

ExampleFind lim

x→0sin

(π

x

)ifitexists.


I f(x) = 1 when x =1

2k + 1/2foranyinteger k

I f(x) = −1 when x =1

2k− 1/2foranyinteger k

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Weird, wildstuffcontinued

Hereisagraphofthefunction:

. .x

.y

..−1

..1

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

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Whatcouldgowrong?SummaryofLimitPathologies

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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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.

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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

.a

.L

lesson03 the concept of limit 027 slides

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