整理02第二章 極限與連續(xù)

1..2.sin口，口101(代表同一變量口)e口口03.⑴⑵⑶⑷0⑸0⑹⑺⑻.4.數(shù)的性質(zhì).1處f(x)limf(x)Ax0xx0是否一定有定義?解析存在與在處是否有定義無f(x)limf(x)Ax0xx0sinxsinx=在lim1f(x)x0xx在.x0limx0f(x)x2x0limf(x)2x0xx0在點(diǎn)有定義.f(x)x02若和是limg(x)f(x)Alimg(x)limf(x)xx0xx0xx0·=·？limflimlimxxxxxx000.例如2·limfA與limlimxxxxxx0001x1.又因?yàn)橹挥性?limx均存在的條件下，才有x0x0x0與limlimxxxx00·=··limflimflimlimxxxxx·xxx0000·=．limflimlimxxxxxx000問題3是否正確，為什么?1xx0解析不正確.盡管，而1limexx0110.1limelimelimxx1x0x0x0ex這說明,不是無窮大.1x0ex例1求下列極限:)；lim(x2x(cosx)22tanxπx42x3；x12x1x1x；1xx13x1；xxsin)x0；limsin(x2x)x3x5.lim1xsinx2x解(1)在π處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．為01limxsin0x0xxx1111為x0時的無窮小.sinxsinxxx(5)limsin(x2x)x(函數(shù)符號與極限符號交換)sinlim(x2x)x(x2x)(x2x)sinlimx2xx2x2xxsin00．3x5xx21x(3x5)xlim11(sin)xxxxxx11)xxx5lim(3)1sin口lim1）x11口lim)0xx10x31例2設(shè)sin,0,問存x2xf(x)alimf(x)x,x0,xx0a2在，并求此極限值.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設(shè)是x0x2f(x)aaax,x0,x的間斷點(diǎn)?是什么間斷點(diǎn)?f(x)解aaxf(x)xx0x0(aax)(aax)limx(aax)x0xx0x(aax)1axx0a1，2acosx1，limf(x)limx22x0x0當(dāng)是的x011a1f(x)f(x)f(x22ax0x00af(0)與f(0)為的跳躍間斷點(diǎn).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,在數(shù)f()x0x0⑵分段函數(shù)必有間斷點(diǎn).(×),x0,f(),x0，是limf(x)0；limf(x)limx0limf(x)limx0x0x0x0x0x0f(0)0limf(x)f(0)在f(x)x0x0⑶與是時的等價無窮小.tan3xsin3xx0(√)tan3x11tan3xlimlimsin3xcos3xx0x0與是x0sin3x⑷無界函數(shù)不一定是無窮大量.(√)，f(x)xx當(dāng)π，2（x2nπxnxf(x)xx02.⑴下列極限存在的是(B);;x3;1lim4xx3x13xxx01.limsinx1x1，，lim4xlim40lim4xxxxx11x1133x31limlim3x13xx3x3xxx0x011x0，limsinx1x1x1x1ax5x1,(Calim6x1；5；6；5a1ax5a6．x1lima6x1xx⑶1在處(Cx0f(x)2x右極限存在.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，知函數(shù)的定義區(qū)間為2320f(x)xx,12,．f(x)x3x2,12,2sinx是x0的間斷點(diǎn);f(x)xsinxf(x)sinx解x01limxxx0sinx是x0f(x)x(4)若(為常數(shù))lim(x)aa．lime(x)eaxx解．lime(x)e()exaxx4.⑴；1cossinlim0解一2sinsin21112．1cossin22limlim1limlim2cos20sincos002cos02222解二無窮小量的等價代換，由于時，02，~,1cos~221．1cossin2limlim200⑵設(shè)；f(x)f(x)xlimx1x1解由無窮小量的等價代換，即時，x0x1，f(x)x1x1~x11．f(x)lnxx1limlimlimx1x1x1x1x1x1⑶解；limesinxxxsinxxex．limesinx0xx,f()6xx1,x1,⑷設(shè)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設(shè)求,在處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