文登考研数学--高等数学--习题集及其答案.docx

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第一章 函数·极限·连续

一.填空题

已知

f(x)?sinx, f[?(x)]?1?x2,则?(x)???,定义域为 .

解. f[?(x)]?sin?(x)?1?x2, ?(x)?arcsin(1?x2)

2?1?1?x2?1,0?x2?2, |x|?

2

?1?x?ax a

设lim?

? ?? tetdt,则a= .

x??? x ? ??

a

解.可得ea??atetdt=(tet?et)

?? ??

?aea?ea

,所以a=2.

3.lim? 1 ? 2 ??? n ?= .

?

?n???n2

?

?n?1 n2

?n?2 n2

?n?n?

?解.1 2 ???

?

解.

n2?n?n n2?n?n n2?n?n

1 ?

2 ??? n 1 2 ??? n

?n2?n?1 n2

?

?n?2 n2?n?n n2

?n?1 n2

?n?1 n2

?n?1

所以 1?2???

所以

1 ? 2

??? n

1?2???n

n2?n?n n2?n?1 n2?n?2 n2?n?n n2?n?1

n(1?n)

1?2???n

???2 ?

1

,(n??)

n2?n?n n2?n?n 2

1?2???n?

n(1?n)

2

1? ,(n??)

1

n2?n?1 n2?n?1 2

??1所以 lim ?

?

?

1

2 ???

n ? 1

?=

n???n2

?n?1 n2

?n?2 n2

?n?n? 2

?0?1 |x|?

?0

已知函数

f(x)?? |x|?1,则f[f(x)] .

解.f[f(x)]=1.

n?

n?3 n

n? n

n??

n?

n?3 n

)= .

n? n

n? n

( n?3 n? n?

n?3

n)( n?3 n?

n? n)

n?

n?

n

解.

nn?? n??

n

=lim

n??

n?3

n?

?2

nn?3

n

n?3 n? n? n

设当x?0时,f(x)?ex?1?bx为x的3阶无穷小,则a???,b???.

解.k?lim

ex?

1?ax

1?bx?limex

bxex

?1?ax?limex

bxex

?1?ax

x?0 x3

x?0

x3(1?bx)

x?0 x3

?limex?bex?bxex?a

x?0 3x2

?limex?2bex?bxexx?0 6x2

(1)

(2)

由(1):

由(2):

lim(ex?bex?bxex?a)?1?b?a?0

x?0

lim(ex?2bex?bxex)?1?2b?0

x?0

1 1

b?? , a?

2 2

limcotx? 1 ?1?

?sinx x?= .

x?0 ?

解.limcosx?

?

x?sinx

?lim

x?sinx

?lim1?cosx

?limsinx?1

x?0sinx

xsinx

x?0 x3

x?0

3x2

x?0

6x 6

lim

n1990

?A

已知

n??nk

?(n?1)k

(?0??),则A=

,k= .

解.lim

n1990

?lim

n1990 ?A

n??nk

?(n?1)k n??knk?1?

所以 k－1=1990, k=1991;

1?A，A?1? 1

k k 1991

二.选择题

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

?(x)

(a)?[f(x)]必有间断点(b)[