View
5
Download
0
Category
Preview:
Citation preview
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( )
Recommended