1、Section 9.2Fermat2Limit of a Function of Two Variableslies close to a fixed real If the values of a real-valued function(,)f x y00(,),xy(,)x yapproaches but not equal towe say that a is the limit of f as 00(,).xyIn symbols,we write00(,)(,)lim(,),x yxyf x yA sufficiently close to the point number A f
2、or all points(,)x y00(,)xy(,)x y00(,)xyequals A.”and we say,“The limit of f as approaches This is like the limit of a function of one variable,except that two independent variables are involved instead of one,complicating the issue of“closeness.”3Limit of a Function of Two Variablesfrom any directio
3、n,whereas in the single-variable case,x onlyalong the x axis.0 xapproached can approach(,)x y is an interior point of fs domain,00(,)xyIf 00(,)xyDefinition(Limit of a Function of Two Independent Variables)(,)x y positive number,there is a positive number such that for all in the domain of f,2200|(,)
4、|,0()().f x yA holds for allxxyy We write00(,)(,)lim(,),x yxyf x yA if,given any The function f has limit A as approaches(,)x y00(,)xy00(,)xyis an accumulation point of the domain of f.and assumeLimit of a Function of Two Variables222(,)(0,0)lim0.x yx yxy Utilize the definition to prove ExampleProof
5、The domain of the function 222(,)x yf x yxy is 2(0,0),AR and 22222220.1x yyyxyxyyx we have0,Then220(0)(0),xy such that if 2220.x yxy By the definition of the limit,we obtain the conclusion.45Limit of a Function of Two Variables Show thatExample 2222001lim()sin0.xyxyxy ProofSince 22221()sin0 xyxy 222
6、21sinxyxy 22,xywe have0,then220(0)(0),xy such that if 22221()sin0.xyxy By the definition of the limit,we obtain the conclusion.0.850.90.9511.051.11.151.21.256Limit of a Function of Two VariablesFind the limit of Example 2200lim().xyxyxy Solution(continued)By the squeeze theoremThen2200limln()0 xyxyx
7、y22000lim()1.xyxyxyeFinish.7Limit of a Function of Two Variablesdoes not exist.Example 36200limxyx yxy Prove thatit is easy to see that the“limit”may be different36200limxyx yxy 3336260limxy kxxkxxk x 2,1kk Proof3,ykx thenIf we takewith different k.Then by the definition of the limit,we obtain theco
8、nclusion.8Continuity of a Function of Two VariablesDefinition (Continuity at a Point,Continuity)ifA function(,)f x yis continuous at the point 00(,)xy00(,)xy1.f is defined at 0000(,)(,)lim(,)(,).x yxyf x yf xy 3.exists00(,)(,)lim(,)x yxyf x y2.A function is continuous if it is continuous at every po
9、int of its domain.As with the definition of limit,the definition of continuity applies atboundary points as well as interior points of the domain of f.The only remain in the domain at all times.(,)x yrequirement is that the point And you can guess or prove that the sums,differences,of continuousfunc
10、tions are continuous where defined.9Continuity of a Function of Two VariablesExample Discuss the continuity of the following function2222(,).xyf x yxy Solution2222(,)xyf x yxy is a product of 221.vxy and22uxy22uxy The function2(,)(0,0).Ax yRSince2222(,)xyf x yxy is a continuous function,221vxy is a
11、continuous function on the plane R2 expect at the point(0,0).functionis continuous in its domain The point(0,0)is a discontinuous point.Finish.Thus the10Continuity of a Function of Two VariablesExampleShow that222222,0(,)0,0 xyxyxyf x yxy is at the origin.Proof,ykx thenIf we take2200limxyxyxy 22220l
12、imxy kxkxxk x 21kk it is easy to see that the“limit”may bedifferent with different k.Then by the definition of the continuity,we obtain the conclusion.11Continuity of a Function of Two VariablesTheorem Suppose that 2RA is a closed bounded region,and:RfA is a continuous function.Then(1)(Boundedness)f
13、 is bounded on the region A.(2)(Maximum and Minimum theorem)f attains both amaximum value and a minimum value in the region A.Theorem(Intermediate value theorem)Let m and M be theminimum and maximum value of the function f in the region A.Forany constant,m M,there must exist at least one point00(,)xyA such that 00(,).f xy
