LimitsandContinuity(section3)

ToDo:

-  Readsection3intheFunctionsofSeveralVariablesmodule

-  Completethissetofnotesasyouwatchthesection3videopostedinTeams(B.LectureContent>Videos)

-  WorkonrelevantAssignmentsandSuggestedPracticeProblemspostedonthewebpageundertheSCHEDULE+HOMEWORKlink

-  PostintheAvenuetoLearnDiscussionForum

LimitofaFunctioninR

Definition:meansthatthey-valuescanbemadearbitrarilyclose(ascloseaswe’dlike)toLbytakingthex-valuessufficientlyclosetoa,fromeithersideofa,butnotequaltoa.

y=f(x)

a

L

x

y

€

limx→a

f (x) = L

2

ExistenceofaLimitinR

Thelimitexistsifandonlyiftheleftandrightlimitsbothexist(equalarealnumber)andarethesamevalue.

2

ExistenceofaLimitinR

Example:**Pleaseworkthroughthesereviewexamplesonyourown.**

Evaluatethefollowinglimitsorshowthattheydonotexist.

(a)(b)(c)

2

€

limx→0

xx

€

limx→1

f (x) where f (x) =x when x <11x 2 when x ≥1

$

% &

' &

€

limx→0

1x 2

ExistenceofaLimitinR

Itisrelativelyeasytoshowthatthistypeoflimitexistssincethereareonlytwowaystoapproachthenumberaalongtherealnumberline:eitherfromtheleftorfromtheright

A XX

2

LimitofaFunctioninR

Definition:meansthatthez-valuesapproachLas(x,y)approaches(a,b)alongeverypathinthedomainoff.

€

lim(x,y )→(a,b )

f (x,y) = L

3

ExistenceofaLimitinR

Ingeneral,itisdifficulttoshowthatsuchalimitexistsbecausewehavetoconsiderthelimitalongallpossiblepathsto(a,b).

3

ExistenceofaLimitinR

However,toshowthatalimitdoesn’texist,allwehavetodoistofindtwodifferentpathsleadingto(a,b)suchthatthelimitofthefunctionalongeachpathisdifferent(ordoesnotexist).

3

ExistenceofaLimitinR

Example:Showthatthefollowinglimitsdonotexist.(a)

3

lim(x , y )→(0,0)

y2 − x2

2x2 +3y2

ExistenceofaLimitinR

Example:Showthatthefollowinglimitsdonotexist.**Pleaseworkthroughthisexampleonyourown.Wewilldiscussittogetherduringourlivesession**

(b)

3

lim(x , y )→(0,0)

6x3y2x4 + y4

ExistenceofaLimitinR

Example:Showthatthefollowinglimitsdonotexist.**Pleaseworkthroughthisexampleonyourown.Wewilldiscussittogetherduringourlivesession**

(c)

Hint:YouwillneedtouseL’Hopital’sRule!Also,usingatrigonometricidentitywillhelpsimplifytheprocess!

3

€

lim(x,y )→(0,0)

x 2 + sin2 y2x 2 + y 2

LimitLaws

Theorem:Assumethatandexist(i.e.arerealnumbers).Then(a)(b)

€

lim(x,y )→(a,b )

f (x,y)

€

lim(x,y )→(a,b )

g(x,y)

€

lim(x,y )→(a,b )

f (x,y) ± g(x,y)( ) = lim(x,y )→(a,b )

f (x,y) ± lim(x,y )→(a,b )

g(x,y)

€

lim(x,y )→(a,b )

c f (x,y)( ) = c lim(x,y )→(a,b )

f (x,y), where c is any constant.

LimitLaws

Theorem(continued):(c)(d)

€

lim(x,y )→(a,b )

f (x,y) × g(x,y)( ) = lim(x,y )→(a,b )

f (x,y) × lim(x,y )→(a,b )

g(x,y)

€

lim(x,y )→(a,b )

f (x,y)g(x,y)

=lim

(x,y )→(a,b )f (x,y)

lim(x,y )→(a,b )

g(x,y), provided lim

(x,y )→(a,b )g(x,y) ≠ 0.

SomeBasicRulesForthefunctionForthefunctionForthefunction.

€

lim(x,y )→(a,b )

f (x,y) = lim(x,y )→(a,b )

x = a

€

lim(x,y )→(a,b )

f (x,y) = lim(x,y )→(a,b )

y = b

€

lim(x,y )→(a,b )

f (x,y) = lim(x,y )→(a,b )

c = c

€

f (x,y) = x,

€

f (x,y) = y,

€

f (x,y) = c,

EvaluatingLimits

Example#10:Usingthepropertiesoflimits,evaluate

Solution:

lim(x , y )→(2,−2)

1xy − 4

.

lim(x , y )→(2,−2)

1xy − 4

=lim

(x , y )→(2,−2)1

lim(x , y )→(2,−2)

xy − 4( )

=lim

(x , y )→(2,−2)1

lim(x , y )→(2,−2)

x ⋅ lim(x , y )→(2,−2)

y − lim(x , y )→(2,−2)

4

=1

2 ⋅ (−2)− 4

= −18

DirectSubstitutionTheorem:Ifisapolynomialorrationalfunction(inwhichcasemustbeinthedomainof),then. €

lim(x,y )→(a,b )

f (x,y) = f (a,b)

€

€

f (x,y)

€

f

€

(a,b)

ContinuityofaFunctioninR

Intuitiveidea:Afunctioniscontinuousifitsgraphhasnoholes,gaps,jumps,ortears.Acontinuousfunctionhasthepropertythatasmallchangeintheinputproducesasmallchangeintheoutput.

3

ContinuityofaFunctioninR

Definition:Afunctioniscontinuousatthepointif

€

lim(x,y )→(a,b )

f (x,y) = f (a,b)

3

€

f

€

(a,b)

ContinuityofaFunctioninR

Example:**Pleaseworkthroughthisexampleonyourown.Wewilldiscussittogetherduringourlivesession**Determinewhetherornotthefunctioniscontinuousat(0,0).

€

f (x,y) =x 2 + y 2 + 4 if (x,y) ≠ (0,0)1 if (x,y) = (0,0)

# $ %

3

WhichFunctionsAreContinuous?

BasicContinuousFunctions:ü polynomialsü  rationalfunctionsü exponentialfunctions

ü  logarithmicfunctions

ü  trigonometricfunctionsü  rootfunctions

Afunctioniscontinuousifitiscontinuousateverypointinitsdomain.

WhichFunctionsAreContinuous?CombiningContinuousFunctions:Thesum,difference,product,quotient,andcompositionofcontinuousfunctionsiscontinuouswheredefined.Example:Findthelargestdomainonwhichiscontinuous.

€

f (x,y) = ex2y + x + y 2

LimitsofContinuousFunctions

Bythedefinitionofcontinuity,ifafunctioniscontinuousatapoint,thenwecanevaluatethelimitsimplybydirectsubstitution.Example:**Pleaseworkthroughthisexampleonyourown.Wewilldiscussittogetherduringourlivesession**

Evaluate

lim(x,y)→(0,−1)

ex2y + x + y2( )