(1)limx2sin1?
x?0x 解 limx2sin1?0(当x?0时? x2是无穷小? 而sin1是有界变量)?
x?0xx (2)limarctanx?
x??x 解 limarctanx?lim1?arctanx?0(当x??时? 1是无穷小?
x??xx??xx而arctan x是有界变量)?
4? 证明本节定理3中的(2)?
习题1?5
1? 计算下列极限?
(1)limx2?5x?2x?3? 解 limx2?5x?2x?3?22?52?3??9? (2)x2xlim?3?3x2?1? 解 x2?3(3)2?3xlim?3x2?1?(3)2?1?0? (3)limx2?2x?1x?1x2?1? 解 limx2?2x?1x?1x2?1?lim(x?1)2x?1(x?1)(x?1)?limx?1x?1x?1?02?0? (4)lim4x3?2x2?xx?03x2?2x? 解 lim4x3?2x2?xx?03x2?2x?lim4x2?2x?1x?03x?2?12? (5)lim(x?h)2?x2h?0h?
222(x?h)2?x2x?2hx?h?x 解 lim?lim?lim(2x?h)?2x?
h?0h?0h?0hh (6)lim(2?1?1)?
x??xx21?lim1?2? 解 lim(2?1?1)?2?limx??x??xx??x2xx22x (7)lim2?1? x??2x?x?11?122?1?limx?1? 解 limx2x??2x?x?1x??2?1?122xx2xx? (8)lim4?2x??x?3x?12x?0(分子次数低于分母次数? 极限为零)? 解 lim4x?2x??x?3x?11?1223x?xxxlim?lim?0? 或
x??x4?3x2?1x??211?2?4xx2 (9)limx2?6x?8?
x?4x?5x?42x?6x?8?lim(x?2)(x?4)?limx?2?4?2?2?
解 lim2x?4x?5x?4x?4(x?1)(x?4)x?4x?14?13 (10)lim(1?1)(2?1)? 2x??xx1)?lim(2?1)?1?2?2? 解 lim(1?1)(2?1)?lim(1?x??xx2x??xx??x2 (11)lim(1?1?1? ? ? ? ?1)?
n??242n1?(1)n?12)?lim?2? 解 lim(1?1?1? ? ? ? ?1nn??n??12421?2 (12)lim1?2?3? ? ? ? ?(n?1)? 2n??n(n?1)n1?2?3? ? ? ? ?(n?1)2?1limn?1?1? ?lim 解 limn??n??2n??n2n2n2(n?1)(n?2)(n?3)?
n??5n3(n?1)(n?2)(n?3)1 解 lim? (分子与分母的次数相同? 极限为
n??55n3最高次项系数之比)?
(n?1)(n?2)(n?3)11)(1?2)(1?3)?1? 或 lim?lim(1?n??5n??nnn55n3 (14)lim(1?33)?
x?11?x1?x (13)lim2131?x?x?3??lim(1?x)(x?2) ?)?lim 解 lim(x?11?x1?x3x?1(1?x)(1?x?x2)x?1(1?x)(1?x?x2)x?2??1? x?11?x?x2 2? 计算下列极限? ??lim32x?2x (1)lim? x?2(x?2)232(x?2)20x?2x??? 解 因为lim3??0? 所以limx?2(x?2)2x?2x?2x2162x (2)lim? x??2x?12x 解 lim?? (因为分子次数高于分母次数)? x??2x?1 (3)lim(2x3?x?1)?
x?? 解 lim(2x3?x?1)??(因为分子次数高于分母次数)?
x?? 3? 计算下列极限? (1)limx2sin1?
x?0x 解 limx2sin1?0(当x?0时? x2是无穷小? 而sin1是有界变量)?
x?0xx (2)limarctanx?
x??x 解 limarctanx?lim1?arctanx?0(当x??时? 1是无穷小?
x??x??xxx而arctan x是有界变量)?
4? 证明本节定理3中的(2)?
习题 1?7
1? 当x?0时? 2x?x2 与x2?x3相比? 哪一个是高阶无穷小？
232x?xx?x 解 因为lim?lim?0? x?02x?x2x?02?x所以当x?0时? x2?x3是高阶无穷小? 即x2?x3?o(2x?x2)?
2? 当x?1时? 无穷小1?x和(1)1?x3? (2)1(1?x2)是否同阶？是否等价？
23(1?x)(1?x?x2)1?x 解 (1)因为lim?lim?lim(1?x?x2)?3? x?11?xx?1x?11?x所以当x?1时? 1?x和1?x3是同阶的无穷小? 但不是等价无穷小?
1(1?x2)?1lim(1?x)?1? (2)因为lim2x?11?x2x?1所以当x?1时? 1?x和1(1?x2)是同阶的无穷小? 而且是等价无穷小?
2 3? 证明? 当x?0时? 有? (1) arctan x~x?
2x (2)secx?1~? 2y?1(提示? 令y?arctan x? 则当x?0时? 证明 (1)因为limarctanx?limx?0y?0tanyxy?0)?
所以当x?0时? arctanx~x?
2sin2x2sinxcosx?lim2?lim(2)2?1? (2)因为limsecx?1?2lim1?22x?012x?0xcosxx?0x?0xxx2222x所以当x?0时? secx?1~? 2 4? 利用等价无穷小的性质? 求下列极限? (1)limtan3x?
x?02xsin(xn) (2)lim(n? m为正整数)?
x?0(sinx)msinx? (3)limtanx?x?0sin3x (4)limsinx?tanx? x?0(31?x2?1)(1?sinx?1) 解 (1)limtan3x?lim3x?3?
x?02xx?02x21 n?mn??sin(xn)x?lim??0 n?m? (2)limx?0(sinx)mx?0xm??? n?m1x2sinx(1?1)sinx?lim1?cosx?lim21? cosx?lim? (3)limtanx?x?0x?0x?0cosxsin2xx?0x2cosx2sin3xsin3x (4)因为
sinx?tanx?tanx(cosx?1)??2tanxsin2x~?2x?(x)2??1x3(x?0)?
2222x1x2(x?0)? 1?x?1?~3(1?x2)2?31?x2?1332 1?sinx?1?sinx~sinx~x(x?0)? 1?sinx?1?1x3sinx?tanx?lim2??3? 所以 lim3x?0(1?x2?1)(1?sinx?1)x?012x?x3
5? 证明无穷小的等价关系具有下列性质? (1) ? ~? (自反性)?
(2) 若? ~?? 则?~?(对称性)? (3)若? ~?? ?~?? 则?~?(传递性)? 证明 (1)lim??1? 所以? ~? ?
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