Ϊlimn??111? lim(n2?n?n)=
n??2211??1n 8 limn??2n?1?1
2n?2?n2n?1?1(2n?1?1)(2n?2?n) ?limn??2n?2?n(2n?2?n)(2n?2?n) limn??4n2?6n?2?2n?2?2n2?n?n ?lim
n??n?24? ?limn??622211?2??2?2??nnnnnn 21?n ?2?2 2͵ȳȱνлм
9 lim
n??
11??1?22?3??1
n(n?1)?11? nn?1
11??1?22?31111?1????223n(n?1) ?1? lim
n??1 n?111??1?22?3?11?lim(1?)=1
n(n?1)n??n?1132n?1?2??n
n??222132n?1 S??2?? ? n2221132n?1 S?2?3??n?1 ?
2222S1222n?1 ?-?ʽ??2??n?n?1
2222212n?1 ʵ S?3?n?2? n22 10 lim
5
2n?1?3
n??n??2n?2n??2n132n?1 lim?2???3 nn??222 limS?3?lim1?lim
1 ùʽlim(?1n??n 11 lim(1?m??n?)eм
1m) m1m1)?lim(1?)?n
m??n??mn1?1 ?[lim(1?)n]?1?e
n??n11 12 lim(1??2)n
n??nn m??n lim(1? lim(1?n??11n?)=lim[(1?n??nnn2n2n?1n?1n?1n2?n)]2
?e
ģùԭм
ͨ˷мתΪޣȻ
?? ԭ:fU(x0;??)ж.limf(x)ڵijҪǣκκU(x0;??)x?x0x0ΪУxnlimf(xn)
x?x013 limnsinn???n
limnsinn???nsin?limx???x???1?0?0 ?xx УnϸϽ. ɹԭ֪limnsinn??n?n?0
a?nbn) a?0,b?0 14 lim(n??2 (1) a,bһΪ0ʱa?0
6
a?nbnb)?limn?0?ab. ? lim(n??n??221ax?bxax?bxx)lny??ln (2) a?0,b?0ʱy?(
x221n1ax?bx2axlna?bxlnb?limx?() limlny?lim?lnx?0x?0xx?0a?bx22 ?lnab nax?bxxa?nbn)?ab ? )?ablim( lim(n??x?0221a?nbn)?ab ??ʽɵlim(n??2
n м
һм.
һЩΪӵУУnеȵ.·еļ.
1õн綨.
ʹñʱ֤Уan1ݼ2н磨Ͻ½磩˶ûֱмֵǸԶнϽ̺ͼޱԵȣм.
1 x1?4n?2ʱ.xn?1xn?1?xn?1limxn
n??2222xn?2?2xn2?xn Ȼxn?0xn?1?xn? ?2xn2xn xn?1xn?1?1xn?1xn?1??2 ?2xn?1227
xn?1?xn?0Уxnݼ½磬ʣxnڣlimxn?xn??
ɵƹϵʽx?
1x?x?2.limxn=2n??x2
2canc2 0?c?1,a1? an?1??Уan
222[4] ѧɷ֤ o?an?1 n?1,2,3 ѧɷ֤
an?1?an n?1,2,3 Ȼa2?a1ak?ak?1 ak?1?ak?. ? . ?
1122ak?ak?1=ak?ak?1ak?ak?1?0.
22n?? Ӷ?. ??֪anϽ磬lim?lڣ
cl2 l??l?1liman?l?1?1?c n??22 2 stolzȥм.
ô˷ʱҪĽһвֳΪУanbnȻжУanbnǷstolzø÷ȥһЩΪn͵ļ.
1k?2k??nk 3 мlimkΪȻ. k?1n??n an?1k?2k??nk,bn?nk?1ɶ2ɵ
anan?1?an(n?1)k?lim lim=lim
n??bn??bn??(n?1)k?nk?1?bnn?1nk(n?1)1 ?lim?n????k?1?nk??k?18