x2sin设limf(x)?A，求证：limf(x)?A． 求极限limx?0sinxx?x0x?x01求极限limx1?sin． 求极限lim?cosln(1?x)?coslnx? x?0xx???1x 111arctan． 求极限lim 求极限limarctanx?arcsin 2x??x??x(1?ex)xx??1?xx2x?1求极限lim． 1x?0求数列的极限lim(sinn?1?sinn) n??2?2x求极限lim 2x设lim?(x)?u0，且?(x)?u0，又limf(u)?Ax?x0u?u0试证：limf??(x)??Ax?x0 设f(x)?x?1lnx试确定实数a，b之值，使得： 当x?a时，f(x)为无穷小；当x?b时，f(x)为无穷大。x设f(x)?，问：当x趋于何值时，f(x)为无穷小。 xtan2 若limf(x)?A，limg(x)?B，且B?Ax?x0x?x0证明：存在点x0的某去心邻域，使得在该邻域内 g(x)?f(x)． 设limf(x)?A，试证明：x?x0对任意给定的??0，必存在正数?，使得对适含不等式0?x1?x0??；0?x2?x0??的一切x1、x2，都有f(x2)?f(x1)??成立。 已知：limf(x)?A?0，试用极限定义证明：limx?x0若数列?xn?与?yn?同发散，试问数列?xn?yn?是否也必发散？ x?x0f(x)?A． x2n?1?x求f(x)?lim2n的表达式 n??x?1 .
2n??x2n?1 (其中a、b为常数，0?a?2?)，设f(x)?lim(1)求f(x)的表达式；x2n?1sin?x?cos(a?bx) (2)确定a，b之值，使limf(x)?f(1)，limf(x)?f(?1)．x?1x??11xn?2?x?n?1的表达式 求f(x)?lim的表达式．n?nn??1?(lnx2)2n?1n??x?x22 设?(x)?x?3x?3，fn(x)?1??(x)??(x)????n(x)，求f(x)?limfn(x)．求f(x)?limn????xxx求f(x)?lim?x?????的表达式． 2222n?1?n??(1?x)(1?x)??1?xnkxn设Sn??，其中bk?(k?1)!，求limSn． 求f(x)?lim的表达式． nn??bn??1?xkk?1?x(1?x)x2(1?x)2xn(1?x)n?求f(x)?lim?1??????的表达式。 2nn??222??求f(x)?limx(1?x)n?x?1(1?x)?1nn?? 的表达式，其中x?0．3an?2(?b)n求数列的极限lim． (其中a?b?0)． n??3an?1?2(?b)n?15?3n?3?(?2)n1352n?1求数列的极限lim． 求数列的极限lim(?????)． nnn??n??32432 求数列的极限lim(1?2q?3q2???nqn?1)，其中q?1．n?? 求数列的极限111??lim????? ?n??a(a?1)(a?2)(a?1)(a?2)(a?3)(a?n?1)(a?n)(a?n?1)??其中a?0．?1?11求数列的极限lim????? ?n??1?33?5(2n?1)(2n?1)???1?111求数列的极限lim??????． ?n??1?22?33?4n(n?1)??a22求数列的极限lim31?22?32???(n?1)2 (其中a?0) n??n1?n2?求数列的极限lim(1?2?3???(n?1)? ?．n??n?2?2????求数列的极限limn(n?2?n?1)． n??求数列的极限limn2?4n?5?(n?1)． n???? .
n4?3n3?6?(n?1)(n?1)求数列的极限lim． n??nan 求数列的极限lim． (其中a?1)．n??2?an10000n111． 求数列的极限lim(1?2)(1?2)?(1?2)． 求数列的极限lim2n??n??n?123nn2?4n?3求数列的极限lim2． 求数列的极限lim(n?1?n)． n??3n?5n?1n??求数列的极限limn1?2n?3n． n??求数列的极限limn??2n?a?2n?1． (a?0，b?0且b?2) n?b?n?232n?1n?1求数列的极限limn(1?)． 求数列的极限limn( ?1)． n??n??2nn?22?10n?3?102n求极限lim． n??3?10n?1?2?102n?1若在x0的某邻域内f(x)?g(x)，且limf(x)?A，limg(x)?B．x?x0x?x0试判定是否可得：A?B． 1 若lim?(x)?0，lim?b?0，则lim?(x)?(x)?0是否成立？为什么？x?x0x?x0?(x)x?x0 确定a，b之值，使limx????3x2?4x?7?(ax?b)?0，x???并在确定好a，b后求极限limx求极限lim(x??3x?4x?7?(ax?b)2? x?12x?cosx?x)． 求极限lim． x??x??3x?sinxx?1(x?1)2?(2x?1)2?(3x?1)2???(10x?1)2求极限lim x??(10x?1)(11x?1)x???求极限limx?x2?2x?5?(x?1)． 求极限lim(4x2?8x?5?2x?1)． x????(x?1)(2x?1)(3x?1)(4x?1)(5x?1)2e3x?3e?2x求极限lim． 讨论极限lim3x． 32?2xx??x??4e(2x?3)(3x?2)?e(x?1)(22x2?1)(32x2?1)(42x2?1)(52x2?1)求极限lim． 335x??(5x?3)?2x(4x2?3)3(3x?2)4a求极限lim． 求极限lim (a?0，a?1)． 25x??x???1?a2x(6x?7)求极限limtan2x?tan(x??4??x)． 4确定a，b之值，使当x???时，f(x)?x2?4x?5?(ax?b)为无穷小． 33x?2?2x3?3x?2x2?5x?6求极限lim． 求极限lim4． 求极限lim． 2x?2x?1x?4x?3x?2x?2x?4 .
2x5x?1?2x?5求极限lim． 求极限lim． 2x?0x?2x?5?5x?42324(1?2x)3?(1?x)5(1?x)?(1?x)求极限lim 求极限lim． x?0(1?4x)2?(1?3x)3?2x?0x2(1?2x)5?(1?4x)3(2x?a)m?am求极限lim (m，n为自然数)． 求极限lim nnx?ax?0xx?a(1?3x)4?1求极限lim． x?0x2ax2?(a?2)x?1设f(x)?2 ax?(a2?1)x?a问：(1)当a为何值时，limf(x)??；x?11； x?12 (3)当a为何值时，limf(x)?0，并求出此极限值。1 (2)当a为何值时，limf(x)?x?2cscx?cotx1?cosax． 求极限lim． x?0x?0xx21?tanx?sinx?1tanx?tan??求极限lim． 求极限lim (0???) x?0x??x3x??21?sinx?cosx求极限lim (p为常数，p?0)． 讨论极限lim2?2cosx． x?01?sinpx?cospxx?0x1?xsinx?cosx求极限limln(1?3x)． 求极限lim． x?0x?0xtanxxen?1?求数列的极限lim(arctan?)n2?1． 求数列的极限limnsin． n??n??nn4??2求数列的极限lim2nsinn?1． 求数列的极限limn(1?cos)． n??n??n2求极限lim 设f(x)是定义在?a，b?上的单调增函数，x0?(a，b)，则(A)f(x0?0)存在，但f(x0?0)不一定存在(B)f(x0?0)存在，但f(x0?0)不一定存在(C)f(x0?0)，f(x0?0)都存在，而limf(x)不一定存在x?x0 (D)limf(x)存在x?x0 答（ ）设x1?a?0，且xn?1? axn，证明：limxn存在，并求出此极限值．n?? 设x1?2，且xn?1?2?xn，证明limxn存在，并求出此极限值。n?? .
设x1?0，且xn?1?n??1a(xn?)(其中a?0)，2xn 证明极限limxn存在，并求出此极限值． x0xn，?，xn?1?1?．1?x01?xn 证明极限limxn存在，并求出此极限值。设x0?1，x1?1?n??111????，(n为正整数) 求证：limxn存在． 222n??23n1111设xn???2???n，求证：limxn存在. n??1?13?13?13?1设xn?1? 11?31?3?5?(2n?1)，x2?，?，xn?，22?42?4?6?(2n)1(1)证明：xn?； 2n?1(2)求极限limxn．设x1?n??100x2?10x?1求极限lim3． 2x??x?01.x?0.01x?0.001x设数列?xn?适合n?1?r?1, (r为定数）证明：limxn?0． n??xn． 2nx?求数列的极限lim． cos(x?)3n??n!6n用极限存在的＂夹逼准则＂证明数列的极限limn?0． n??2111 求数列的极限lim(???)．222n??n?1n?2n?n?求极限limtan3x?3tanx?求数列的极限lim3n??n2sinn!． n?12x?ln(2?3e) 求极限lim． ?．x???ln(3?2e3x)??111求数列的极限lim?????n??(n?1)2(n?2)2(2n)2?ln(x6?5x3?7)x?x?x?x求极限lim． 求极限lim． x??ln(x2?3x?4)x???x .
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