整理02第二章 极限与连续

1..2.sin口，口101(代表同一变量口)e口口03.⑴⑵⑶⑷0⑸0⑹⑺⑻.4.数的性质.1处f(x)limf(x)Ax0xx0是否一定有定义?解析存在与在处是否有定义无f(x)limf(x)Ax0xx0sinxsinx=在lim1f(x)x0xx在.x0limx0f(x)x2x0limf(x)2x0xx0在点有定义.f(x)x02若和是limg(x)f(x)Alimg(x)limf(x)xx0xx0xx0·=·？limflimlimxxxxxx000.例如2·limfA与limlimxxxxxx0001x1.又因为只有在0limx均存在的条件下，才有x0x0x0与limlimxxxx00·=··limflimflimlimxxxxx·xxx0000·=．limflimlimxxxxxx000问题3是否正确，为什么?1xx0解析不正确.尽管，而1limexx0110.1limelimelimxx1x0x0x0ex这说明,不是无穷大.1x0ex例1求下列极限:)；lim(x2x(cosx)22tanxπx42x3；x12x1x1x；1xx13x1；xxsin)x0；limsin(x2x)x3x5.lim1xsinx2x解(1)在π处4f(x)x2sin2x(cosx)2xxπx4xxx)]222tanxxπ4ππππ24()))22444π222()()2222π2．1(2)23xlim()x12x1x2x122x1lim()x1x21lim(1)x12x1x1）lim1）口口111)x22lim(112xx1112)x1)21212xxxx11112)x[lim(1212121xxxx21e12．e0lim1x01x3x12xx)1x(x)33lim(1x)x1x)1x(x)3332x)1x(x)332xx)x1子）x11x(x)332lim1xx13．2lim(sinxxsin)1xxx0sinx1limlimxsinxxx0x0101．为01limxsin0x0xxx1111为x0时的无穷小.sinxsinxxx(5)limsin(x2x)x(函数符号与极限符号交换)sinlim(x2x)x(x2x)(x2x)sinlimx2xx2x2xxsin00．3x5xx21x(3x5)xlim11(sin)xxxxxx11)xxx5lim(3)1sin口lim1）x11口lim)0xx10x31例2设sin,0,问存x2xf(x)alimf(x)x,x0,xx0a2在，并求此极限值.解..，1limf(x)limxsin02xx0x0．limf(x)lim(ax)a2x0x0limf(x)0.limf(x)limf(x),即ax0x0x0a0limf(x)0．limf(x)x0x0x,x0,例3设是x0x2f(x)aaax,x0,x的间断点?是什么间断点?f(x)解aaxf(x)xx0x0(aax)(aax)limx(aax)x0xx0x(aax)1axx0a1，2acosx1，limf(x)limx22x0x0当是的x011a1f(x)f(x)f(x22ax0x00af(0)与f(0)为的跳跃间断点.f(0)f(0)x0f(x)1例4的值.ab,x2limaxb0x1x解x21lim(b)x1xa)x(ab)x1b2x1x，0和2xx1a0ab00,即b1.a⒈⑴在在.xf(x)xf(x)00(×),x0,limf(x)limx0，f()x0,x0x0在limf(x)0f(0)1f(x)x0x0,x0,在数f()x0x0⑵分段函数必有间断点.(×),x0,f(),x0，是limf(x)0；limf(x)limx0limf(x)limx0x0x0x0x0x0f(0)0limf(x)f(0)在f(x)x0x0⑶与是时的等价无穷小.tan3xsin3xx0(√)tan3x11tan3xlimlimsin3xcos3xx0x0与是x0sin3x⑷无界函数不一定是无穷大量.(√)，f(x)xx当π，2（x2nπxnxf(x)xx02.⑴下列极限存在的是(B);;x3;1lim4xx3x13xxx01.limsinx1x1，，lim4xlim40lim4xxxxx11x1133x31limlim3x13xx3x3xxx0x011x0，limsinx1x1x1x1ax5x1,(Calim6x1；5；6；5a1ax5a6．x1lima6x1xx⑶1在处(Cx0f(x)2x右极限存在.因10f(x)2xx1在x01f(x)20()2fxxxx0x01在x01f(x)2()2fxxxx0x01在x0f(x)2x10x(Df(x)x(A);(B)(D);.(C);1在f(x)xy1xxO10xf(x)x3.2(1),b0，a23ba2x1x6；解3xx，a02b2axbx2bx22x1b．b62x1limlimlim32x12xxxx；f(x)x3x2,12,2解由，知函数的定义区间为2320f(x)xx,12,．f(x)x3x2,12,2sinx是x0的间断点;f(x)xsinxf(x)sinx解x01limxxx0sinx是x0f(x)x(4)若(为常数)lim(x)aa．lime(x)eaxx解．lime(x)e()exaxx4.⑴；1cossinlim0解一2sinsin21112．1cossin22limlim1limlim2cos20sincos002cos02222解二无穷小量的等价代换，由于时，02，~,1cos~221．1cossin2limlim200⑵设；f(x)f(x)xlimx1x1解由无穷小量的等价代换，即时，x0x1，f(x)x1x1~x11．f(x)lnxx1limlimlimx1x1x1x1x1x1⑶解；limesinxxxsinxxex．limesinx0xx,f()6xx1,x1,⑷设在x1f(x)f(x)解，limf(x)1．limf(x)limx1limf(x)lim6x51xxxxx1且f1limf(x)f在f(x)x1x1x1x1f(x)xf(x)6x5,．f(x)e,x0,x⑸设求,在处x0f()x0,(x)ff(xf(x)sinxx0x0,x0,xsinx解，，所以limf(x)lime1fxlim()lim1xxx0x0x0x0limf(x)1．x0且f(0)1limf(x)f(0)在f(x)x0x01⑹e1xf(x)1e1x解x0x0111e1，e11e1，xxxf(x)1f(x)111x0x0e1x0x0e1x01exxx1e1即