文档介绍：1数学分析复习(二)多元函数的极限与连续一、多元函数的极限定义设D?Rn,f:D→∈Rn是D的一个聚点(a∈D′),s∈??>0,??>0,当x?D及则称函数f在点a处有(重)极限,或当x趋于a时,f(x)趋于s,记作s)x(flim~ě??或).ax(s)x(f??|f(x)-s|<?,时有????||||0ax2?定义设D?Rn为函数f的定义域,P0为D的一个聚点。如果?M>0,?P0的一个?空心邻域使当P∈),(0?PU?∩D时,M)P(f?,则称f在D上当P→P0时,存在非正常极限+∞,记作?????)P(flimMmmIam),(0?PU??????:)(lim0,PfPPDP??????:)(lim0,PfPPDP?:设D?Rn,f:D→(x0,y0)∈Rn是D的一个聚点(P0∈D′),A∈(x,y) ∈D使当,0,0)(lim0,??????????sPfPPDP时有????????||0,||000yyxx???|)(|APf)),((),(lim)(lim000,DyxAyxfAPfyyxxPPDP???????可写作4???????:),(lim)1(),(),(Ayxfyx?????:),(lim)2(),0(),(Ayxfyx,0,0:),(lim)3(),(),(??????????????Myxfyx??????:),(lim)4(),0(),(yxfyx???|),(|Ayxf有时使当,MyMxM??????,,0,0?有时使当,MyxM????????,||0,0,0,0??????|),(|AyxfMyxfyx????),(,时使当??有时使当,MyxM????????,||0,0,0,0???Myxf?),(5性质:(1)四则运算法则(2)归结原理(3)唯一性、局部有界性、局部保号性(3)无穷小量性质6如何求多元函数的极限?(1)由定义求多元函数的极限。?例1证明:(0,0)),(???yxyxlimīě证明:222222||||yxxyxyyxyx???)(41)4()(2222222yxyxyx?????,4,0??????取时有当????)(P,)),((,2222yxPyxyx????例2证明:.311lim3???????yxyyx7:证明1331311????????yyxyyxy)1(4|3|14)3(????????yyxyxy,08,02,0??????????M取,Myx时当????,|3|0??????????22311yxy例3证明:?????yxyx21lim21证明:,0??M|2||1|2|)2()1(2||2|?????????yxyxyx8时当取???????????|2|0,|1|0,041yxMMMMyx14121|2|????此时,Myx??21??????yxyx21lim219(2))(lim00(0,0)),(???????yxyxyxyx112222(0,0)),(?????yxyxlimīě1)11)(1(1)1)((22222222(0,0)),(????????????yxyxyxyxlimīě??21122(0,0)),(??????yxlimīě(经变形后)1011lim00????xyxyyxxyxyxyyx)11(lim00????)11)(11()11(lim00?????????xyxyxyxyyx211)11(lim00????????xyyxyxxyxx????????????221lim0yxxxyxx?????????????????????22021lim22lim2021limexyxxxxyx?????????????????????????