《高數(shù)雙語(yǔ)》課件section 9.2

Section9.2LimitandContinuityofMulti-variableFunctionsFermat2LimitofaFunctionofTwoVariablesliesclosetoafixedrealIfthevaluesofareal-valuedfunctionapproachesbutnotequaltowesaythataisthelimitoffasInsymbols,wewritesufficientlyclosetothepointnumberAforallpointsequalsA.”andwesay,“ThelimitoffasapproachesThisislikethelimitofafunctionofonevariable,exceptthattwoindependentvariablesareinvolvedinsteadofone,

complicatingthe

issueof“closeness.”3LimitofaFunctionofTwoVariablesfromanydirection,whereasinthesingle-variablecase,xonlyalongthex–axis.approachedcanapproachisaninteriorpointoff’sdomain,IfDefinition(LimitofaFunctionofTwoIndependentVariables)positivenumberε,thereisapositivenumberδ

suchthatforallinthedomainoff,Wewriteif,givenanyThefunctionfhaslimitAasapproachesisanaccumulationpointofthedomainoff.andassumeLimitofaFunctionofTwoVariablesUtilizethedefinitiontoproveExampleProofThedomainofthefunctionisandwehaveThensuchthatifBythedefinitionofthelimit,weobtaintheconclusion.45LimitofaFunctionofTwoVariablesShowthatExampleProofSincewehavethensuchthatifBythedefinitionofthelimit,weobtaintheconclusion.6LimitofaFunctionofTwoVariablesFindthelimitofExampleSolution(continued)BythesqueezetheoremThenFinish.7LimitofaFunctionofTwoVariablesdoesnotexist.ExampleProvethatitiseasytoseethatthe“l(fā)imit”maybedifferentProofthenIfwetakewithdifferentk.Thenbythedefinitionofthelimit,weobtaintheconclusion.8ContinuityofaFunctionofTwoVariablesDefinition(ContinuityataPoint,Continuity)ifAfunctioniscontinuousatthepoint

1.

fisdefinedat3.

exists2.

Afunctioniscontinuousifitiscontinuousateverypointofitsdomain.Aswiththedefinitionoflimit,thedefinitionofcontinuityappliesatboundarypointsaswellasinteriorpointsofthedomainoff.Theonlyremaininthedomainatalltimes.requirementisthatthepointAndyoucanguessorprovethatthesums,differences,…,ofcontinuousfunctionsarecontinuouswheredefined.9ContinuityofaFunctionofTwoVariablesExampleDiscussthecontinuityofthefollowingfunctionSolutionisaproductofandThefunctionSinceisacontinuousfunction,isacontinuousfunctionontheplaneR2expectatthepoint(0,0).functioniscontinuousinitsdomainThepoint(0,0)isadiscontinuouspoint.Finish.Thusthe10ContinuityofaFunctionofTwoVariablesExampleShowthatisnotcontinuousattheorigin.ProofthenIfwetakeitiseasytoseethatthe“l(fā)imit”maybedifferentwithdifferentk.Thenbythedefinitionofthecontinuity,weobtaintheconclusion.11ContinuityofaFunctionofTwoVariablesTheoremSupposethatisaclosedboundedregion,andisacontinuousfunction.Then(1)(Boundedness)fisboundedontheregionA.(2)(MaximumandMinimumtheorem)fattainsbothamaximumvalueandaminimumvalueintheregionA.Theorem(Intermediatevaluetheorem)