第一章 函数·极限·连续
一. 填空题 1. 已知解.
f(x)?sinx,f[?(x)]?1?x2,则?(x)?__________, 定义域为___________.
f[?(x)]?sin?(x)?1?x2, ?(x)?arcsin(1?x2)
?1?1?x2?1, 0?x2?2,|x|?2
axa?1?x?t2．设lim??tedt, 则a = ________. ????x???x?a解. 可得ea??tetdt=(tet?et)??a?aea?ea??, 所以 a = 2.
3.
12n??lim?2?2???2?=________. n??n?n?1n?n?2n?n?n??12n ????n2?n?nn2?n?nn2?n?n12n12n <2<2 ?2???2?2???2n?n?1n?n?2n?n?nn?n?1n?n?1n?n?11?2???n1?2???n12n所以 << ????n2?n?nn2?n?1n2?n?2n2?n?nn2?n?1n(1?n)1?2???n12, (n??) ??22n?n?nn?n?n2n(1?n)1?2???n12, (n??) ??n2?n?1n2?n?12解. 所以
12n??1lim?2?2???2?=n??n?n?1n?n?2n?n?n?2?|x|?1?1, 则f[f(x)] _______. f(x)??
|x|?1?0
4. 已知函数
解. f[f(x)] = 1. 5.
lim(n?3n?n?n)=_______.
n??解.
lim(n?3n?n?n)?limn??(n?3n?n?n)(n?3n?n?n)n?3n?n?nn??
=limn?3n?n?nn?3n?n?nn???2
6. 设当x?0时,f(x)?ex?1?ax为x的3阶无穷小, 则a?_____,b?______.
1?bxex?解.
k?limx?01?axxxxx1?bx?lime?bxe?1?ax?lime?bxe?1?ax
x?0x?0x3x3(1?bx)x3
ex?bex?bxex?a?limx?03x2ex?2bex?bxex?limx?06x2x?0 ( 1 )
( 2 )
由( 1 ): 由( 2 ):
lim(ex?bex?bxex?a)?1?b?a?0 lim(ex?2bex?bxex)?1?2b?0
x?0
11b??,a?22
7.
1??1limcotx???=______. x?0?sinxx?解.
limcosxx?sinxx?sinx1?cosxsinx1??lim?lim?lim? 32x?0sinxx?0x?0x?0xsinxx3x6x6n1990?A(? 0 ? ?), 则A = ______, k = _______. 8. 已知limkn??n?(n?1)k解.
n1990n1990limk?limk?1?A n??n?(n?1)kn??kn??所以 k－1=1990, k = 1991;
二. 选择题
111 ?A，A??kk19911. 设f(x)和?(x)在(－?, +?)内有定义, f(x)为连续函数, 且f(x) ? 0, ?(x)有间断点, 则 (a) ?[f(x)]必有间断点 (b) [ ?(x)]2必有间断点 (c) f [?(x)]必有间断点 (d)
?(x)必有间断点 f(x)?1|x|?1解. (a) 反例 ?(x)?? , f(x) = 1, 则?[f(x)]=1
|x|?10?(b) 反例
?(x)??|x|?1?1 , [ ?(x)]2 = 1 ??1|x|?1?1|x|?1(c) 反例 ?(x)?? , f(x) = 1, 则f [?(x)]=1
|x|?10?(d) 反设 g(x) =
?(x)在(－?, +?)内连续, 则?(x) = g(x)f(x) 在(－?, +?)内连续, 矛盾. 所以(d)是答案. f(x)2. 设函数
f(x)?x?tanx?esinx, 则f(x)是
(a) 偶函数 (b) 无界函数 (c) 周期函数 (d) 单调函数
解. (b)是答案. 3. 函数
f(x)?|x|sin(x?2)x(x?1)(x?2)2在下列哪个区间内有界
(a) (－1, 0) (b) (0, 1) (c) (1, 2) (d) (2, 3) 解. limx?1f(x)??,limf(x)??,f(0?)?x?0sin2sin2 ,f(0?)??44 所以在(－1, 0)中有界, (a) 为答案.
x2?1x?1时,函数e的极限 4. 当x?1x?1(a) 等于2 (b) 等于0 (c) 为? (d) 不存在, 但不为?
11???x2?1x?1解. lime?lim(x?1)ex?1??x?1x?1x?1?01x?1?0. (d)为答案.
x?1?05. 极限lim?352n?1?的值是 ????2222?n??12?222?3n?(n?1)???(a) 0 (b) 1 (c) 2 (d) 不存在 解.
?352n?1? lim?2????2222?n??1?222?3n?(n?1)??=lim??11?1111?1?????????lim1??1, 所以(b)为答案. 22222?2?n??12n???223n(n?1)(n?1)????(x?1)95(ax?1)5?8, 则a的值为 6. 设lim250x??(x?1)(a) 1 (b) 2 (c)
58 (d) 均不对
解. 8 =
(x?1)95(ax?1)5(x?1)95/x95(ax?1)5/x5lim=limx??x??(x2?1)50(x2?1)50/x100(1?1/x)95(a?1/x)555a?8, 所以(c)为答案. ?a =lim, 250x??(1?1/x)7. 设lim(x?1)(x?2)(x?3)(x?4)(x?5)???x??(3x?2), 则?, ?的数值为
(a) ? = 1, ? = 解. (c)为答案. 8. 设
111 (b) ? = 5, ? = (c) ? = 5, ? = 5333 (d) 均不对
f(x)?2x?3x?2, 则当x?0时
(a) f(x)是x的等价无穷小 (b) f(x)是x的同阶但非等价无穷小 (c) f(x)比x较低价无穷小 (d) f(x)比x较高价无穷小
解.
2x?3x?22xln2?3xln3lim?ln2?ln3, 所以(b)为答案. =limx?0x?0x1
高数考研习题及答案
