,当,时,设,,(,),,,(,)且lim存在,xxoo011x,x,0 ,,,,1求证:lim,lim(,,xxxx,,,,001
123若当0时,()(1)1与()cos1是等价无穷小,则x,,x,,ax,,x,x,a,
1313A( B( C(, D(,( 2222
答( )
当x0时,下述无穷小中最高阶的是,
22A x B1 cosx C 1x1 D xsinx,,,, 答( )
n2 求limn,,ln(2n,1),ln(2n,1)之值( 求极限lim(,1)nsin(,n,2)(n,,,,,n22xe,1,x11lim的值,_____________ 求极限lim(n,)ln(1,)( 3,0xn,,2nxsinx
a,a,n1n设有数列a,a,a,b (b,a),a,,12n22 求证:limy,lim(a,a)及lima(nn,1nn,,,,,,nnn
2xxnn,1设x,a,x,b((b,a,0) x,,12,2nx,xnn,1 11记:y,,,求limy及limx(nnnn,,n,,xxn,1n
sinx()cos12,,xx求极限之值(lim 2,0xx
设,;且lim()lim()uxAAvxB,,,0xxxx,,00 vxB()试证明:(lim()uxA,xx,0
12limln(1),x,,,(x,1)x,1
( (1 (0 (ln2A,BCD
答( )
sinxxlim(12)x,,,x0
2(1 ( ( (2ABeCeD 答( )
12ux,,xfu,u设()1sin. ()x fu,fux,()1,,()1求:lim及limu(x)之值,并讨论lim的结果(u,1x,0x,0u,ux,1()12x,9 lim的值等于_____________2x,3xx,,6
x,xe,4elim,x,xx,,3e,2e
1A( B(2 C(1 D(不存在3
答:( )
35()()23,,xxlim,8x,,()6,x
1 不存在....ABCD,115323,
答:( )
1020xx(1,2)(1,3)xlim,____________ lim的值等于____________ 215x,,x,xx(1,6)0x,e,e
3431612,,,xxx,x,32求之值(lim 求极限lim(32x,0x,1()xx,5x,x,x,1
已知:,lim()lim()()uxuxvxA,,,,0xxxx,,00 问,为什么,lim()vx,xx,0
5关于极限结论是:lim1x,0x,e3
55ABCD 不存在0 34
答( )
设limf(x),A,limg(x),,,则极限式成立的是x,xx,x00
f(x)A.lim,0x,xg(x)0
g(x)B.lim,, x,xf(x)0
C.limf(x)g(x),,x,x0
()gxDfx,,.lim()x,x0
答( )
xf(x),ecosx,问当x,,,时,f(x)是不是无穷大量(
1limtanx,arctan,x,0x
,,ABCD,.0 .不存在. . . 22 答( )
2arctan()xlim,x,,x
,.0 . .1 .ABCD, 2
答( )
21x,lim,2x,,3x,
A.2 B.,2 C.,2 D.不存在 答( )
3 设f(x),,则f(,0),___________1x2,e
1limarccot,x,0x
,AB,CD.0 . .不存在. . 2 答( )ax,coslim0,则其中,,ax,01ln,x
, 012....ABCD3
答( )
2x,x,,ee3xlim的值等于____________ x,0,1cosx
212,x(cos)lim,x,0 x
ABCD..... 不存在 220,
答:( )
25pxqx,,设,其中、为常数(fx(),pqx,5
问:、各取何值时,;()lim()11pqfx,x,,
20 、各取何值时,;()lim()pqfx,x,,
、各取何值时,(()lim()31pqfx,x,5
222223nnxx,,,22()32x,()()求极限(lim求极限(lim 22nn32x,,x,,()()xx,,,11()23x,
42xABxcx,,,3,,(,1),(,1)已知lim,02x,1 x(,1)
试确定A、B、C之值(
32axbxcxd,,,已知f(x),,满足(1)limf(x),1,(2)limf(x),0(2x,,x,1 x,x,2
试确定常数a,b,c,d之值(
(a,b)x,b已知lim,4,试确定a,b之值( x,13x,1,x,3
1lim(x)0lim :若,,,则,,:上述说法是否正确,为什么,x,xx,x(x),00
当时,是无穷大,且,xxfxgxA,,()lim()0xx,0 证明:当时,也为无穷大(xxfxgx,,()()0
2x1,lim用无穷大定义证明:,,,( 用无穷大定义证明:limlnx,,,(,x,1x1,x,0
1用无穷大定义证明:limtanx,,, lim用无穷大定义证明:,,,( ,x,,0x,1,0x1,2
3 用无穷大定义证明:lim(x,4x),,,(x,,,
用无穷大定义证明:limlogx,,, (其中0,a,1)(a,,,x
若当时,、都是无穷小,xxxx,,,()()0
则当时,下列
表
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示式哪一个不一定是无穷小xx,.0
()()()Axx ,,,
22()()()Bxx , ,,
1()ln()()Cxx ,,,,,,
2()x,()D ()x,
答( )
:当,是无穷小量:是xxx,,()0
:当时,是无穷小量:的xxx,(),0
()A充分但非必要条件
()B必要但非充分条件
()C充分必要条件
()D既非充分条件,亦非必要条件 答( )
:当时,是无穷小:是xxfxA,,()0
::的:lim()fxA,xx,0
()A充分但非必要条件
()B必要但非充分条件
()C充分必要条件
()D既非充分条件,亦非必要条件 答( )
若,,但(lim()lim()()fxgxgx,,,000xxxx,,00
fx() 证明:的充分必要条件是lim,bxx,0gx()
fxbgx()(),, (lim,0xx,0gx()
n 用数列极限的定义证明:lima,0,(其中0,a,1)(,,n
1n 用数列极限的定义证明:lima,1 (0,a,1)(,,n
nn(,2)1用数列极限的定义证明:lim,( 2n,,2n2,5
sinx,1cos(sinx),,(cosx),1lim的值等于___________ 求极限之值( lim23x,0,0,x2ln(1x)x
x2x(sin)11,,xx,,xx(cos,sin),1lim,____________ 求极限之值(lim 23,0xx,0xx3xxx(1,2),1x(1,sin),1lim,_____________ lim,__________ 2,0x,0xxx
12sinx,,x,12xx(cos),1 求极限之值(lim()x,1,,lim,______________ x,,3x,1,0x,,x
设在x的某去心邻域内0,,(x),u(x),,(x),且当x,x时,,(x)~,(x)(00 试证明:当x,x时 ,(x)~u(x)(0
()0()()设当x,x时,,x,,,x,o,x,,,0
,(x),,(x)()~()lim(0),x,x(存在,A, 1x,x()ux0
()(),x,,x1求证:lim,A(x,x()ux0
571312,,,xx()()求之值(lim 2x,0()211x,,
设当,,,,均为无穷小,xxxxxx,,,,,()()()()011
()x,且;,如果()~()()~()limxxxx,A ,,,,11xx,0()x,
11x,()x,()试证明:(lim()lim()11,,,xx1,,,,1,,xx,,xx00
设当,,都是无穷小,且,xxxxxx,,,,,,,()()()()000 ,()x试证明:(1,()~()()xxx,,,,,
(x),设当xx时,(x)与(x)均为无穷小,且(x)~(x);如果limA,,,,,,011,xx(x),0
aa1(x)1,,,,,1(x)1,,,,,1试证明:limlim(, ,,xxxx(x)(x),,00(式中a是正常数)
1用数列极限的定义证明lim,0( n,,n!
设limx,A,且B,A,C(n,,n
B,AA,C试证必有正整数N存在,使当n,N时恒有 ,x,成立(n22
设有两个数列,满足xy,,,,nn
()lim10,x;n,,n ()()2,yMM 为定数(n
lim(),,0试证明:(xynn,,n
12xsinx 求极限lim设,求证:(lim()lim()fxAfxA,, x,0sinxxxxx,,00
1 求极限(limsinx1, 求极限limcosln()cosln1,,xx,,x,0xx,,,
21x11求极限(limarctan 求极限lim 求极限limarctanarcsinx, x2x,,x,,x,,xxe1,()1,xx
x21,求极限(lim 1x,0 求数列的极限lim(sinn,1,sinn)xn,,22,
设lim,(x),u,且,(x),u,又limf(u),A00x,xu,u00 ,,试证:limf,(x),Ax,x0
x,1设fx(),lnx
试确定实数,之值,使得:ab 当时,为无穷小;xafx,()
当时,为无穷大。xbfx,()
x设,问:当趋于何值时,为无穷小。fx(),xfx() xtan2
若limf(x),A,limg(x),B,且B,Ax,xx,x00
证明:存在点x的某去心邻域,使得在该邻域内 g(x),f(x)(0
设,试证明:lim()fxA,xx,0
对任意给定的,必存在正数,使得对适,,,0
含不等式;的一切00,,,,,,xxxx,,1020
xxfxfx、,都有成立。()(),,,1221
已知:limf(x),A,0,试用极限定义证明:limf(x),A( x,xx,x00,,,,,,若数列x与y同发散,试问数列x,y是否也必发散, nnnn
21n,xx,求的表达式fx()lim, 2nn,,1x,
,21n,xxabxsincos(),,2设fx()lim,2nn,,x,1
其中、为常数,,()aba02,,,
()()1求的表达式;fx
()lim()()lim()()211确定,之值,使,(abfxffxf,,,xx,,,11
n,,n,211x,x求的表达式fx()lim, fx, 求()lim的表达式(221n,n,nn,,n,,,(ln)x1x,xn22 设,(x),x,3x,3,f(x),1,,(x),,(x),?,,(x),求f(x),limf(x)(nnn,,
,,xxx求的表达式(fxx()lim,,,,,? ,,22221n,n,,111,x()(),x,x,,
nnkx设S,,其中b,(k,1)!,求limS( 求的表达式(fx()lim, ,nknn,,nbn,,1,xk1,k
nn22,,xxxxxx()()()1,1,1, 求的表达式。fx()lim,,1,,,?,,n2n,,222,,
nx(1,x),x,1 求f(x),lim的表达式,其中x,0(n,,n(1,x),1
nn3a,2(,b)求数列的极限lim( (其中a,b,0)( n,1n,1n,,3a,2(,b)
nn()5332,,,,13521,n求数列的极限(lim 求数列的极限(lim(),,,,? nnn,,n,,324322n,1 求数列的极限lim(1,2q,3q,?,nq),其中q,1(n,,
求数列的极限
111,,,,,?lim ,,n,,aaaaaaananan,,12,,,123,,,,,11()()()()()()()(),,
a,0其中(
,,111lim,,,求数列的极限? ,,n,,1335(2n1)(2n1),,,,,,
,1111,lim求数列的极限,,,?,( ,,n,,122334n(n1),,,,,,
2a2222,, 求数列的极限lim1,2,3,?,(n,1) (其中a,0)3n,,n
2,,1nlim(123(1)求数列的极限,,,?,,,( n,,n,,22,n,,
求数列的极限(lim()nnn,,,21n,,
2 求数列的极限(lim()nnn,,,,451,,n,,
43nnnn,3,6,(,1)(,1)求数列的极限( limn,,n
na 求数列的极限lim( (其中a,1)(n,,n2,a
10000n111求数列的极限(lim 求数列的极限lim(1,)(1,)?(1,)( 2222n,,n,,,1nn232nn,,43求数列的极限(lim 求数列的极限(lim()nn,,12n,,n,,351nn,,
nnn 求数列的极限(lim123,,n,,
2n,a,2n,1求数列的极限lim( (a,0,b,0且b,2) n,,n,b,n,2
321n,n,1 求数列的极限(lim()n1,求数列的极限(lim()n,1n,,n,,2nn,2
nn2210310,,,求极限(lim nn,,121n,,310210,,,
若在x的某邻域内f(x),g(x),且limf(x),A,limg(x),B(0x,xx,x00
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,xx,xx,x,(x)000
2确定,之值,使,abxxaxblim()3470,,,,,,,x,,, 2并在确定好,后求极限abxxxaxblim()347,,,,,,x,,,
,1xcos2xx, 求极限(lim()x,x求极限(lim x,,x,,x,1sin3xx,2222()()()()xxxx,,,,,,,,12131101?求极限lim x,,()()101111xx,,
22 求极限(lim()xxxx,,,,251 求极限(lim()48521xxx,,,,,,x,,,x,,,xx,32()()()()()xxxxx,,,,,12131415123ee,求极限(lim 讨论极限(lim 32xx,32x,,x,,()()2332xx,,4ee,
22222222()()()()()xxxxx,,,,,121314151求极限(lim 335x,,()532x,,
234x4332xx,,()()a求极限(lim ,,01求极限 ,(lim() aa252xx,,x,,,()67x,1,a
, 求极限(limtantan()2xx,,,4x,4
2确定a,b之值,使当x,,,时,f(x),x,4x,5,(ax,b)为无穷小(
323xx,,32xx,,56322x,,求极限(lim 求极限(lim 求极限(lim 42x,1x,2x,2xx,,43x,4x,2
2x5125xx,,,求极限(lim 求极限(lim 2x,0x,2,,x55x,4352324()()121,,,xx()()11,,,xx求极限lim 求极限(lim 232x,0x,0()()14132,,,,xxx
53mm,,()()1214,,,xx(2xa)a求极限lim (m,n为自然数)( 求极限lim nn,x,0xax,xa
4()131,,x求极限(lim x,0x
2221axax,,,()设fx(), 22axaxa,,,()1
问:当为何值时,;()lim()1afx,,x,1
1 当为何值时,;()lim()2afx, x,12
当为何值时,,并求出此极限值。()lim()30afx,1x,2
csccotcosxx,1,ax求极限(lim求极限(lim 2x,0x,0xx
tansin11,,,xxtanx,tan,,求极限(lim 求极限lim (0,,,) 3x,0x,,x,,2x
sincos1,,xxcos22,x求极限 为常数,(lim()pp,0 讨论极限(lim x,0sincosx,01,,pxpxx
ln()13,x1sincosxxx,,求极限(lim lim求极限( x,0x,0tanxxx
en,1,2求数列的极限(limsinn 求数列的极限lim(arctan,)n,1( n,,n,,n4n
,,n2lim2sin求数列的极限(lim(cos)n1, 求数列的极限( n,1n,,n,,n2
设()是定义在,上的单调增
函
关于工期滞后的函关于工程严重滞后的函关于工程进度滞后的回复函关于征求同志党风廉政意见的函关于征求廉洁自律情况的复函
数,fx,,ab
(,),则x,ab0
(A)f(x0)存在,但f(x0)不一定存在,,00
()(0)存在,但(0)不一定存在Bfx,fx, 00
()(,0),(,0)都存在,而lim()不一定存在Cfxfxfx00x,x0
(D)limf(x)存在x,x0
答( )
设x,a,0,且x,ax,证明:limx存在,并求出此极限值(,1n1nn,,n
设x,2,且x,2,x,证明limx存在,并求出此极限值。1n,1nn,,n
1a设,且其中,xxx,,,,0()()a011nn,2xn 证明极限存在,并求出此极限值(limxnn,,
xx0n设,,,(xx,,,11?x,,101n,11,x1,x0n 证明极限存在,并求出此极限值。limxnn,,
111设x,1,,,?,,(n为正整数) 求证:limx存在( nn222,,n23n
1111,,,,?,设x,求证:limx存在. nn2n,,n,,1131,,3131
113,13521,,,?()n设,,,,xxx,,?,12n224,2462,,?()n
1()1证明:;x, n21n,
()lim2求极限(xnn,,
2100101xx,,求极限(lim 32x,,...xxx,,,010010001
xn,1,,设数列x适合,r,1, (r为定数)证明:limx,0( nnn,,xn
3tantanxx,3求极限(lim n,2,x,求数列的极限(lim cos()x,3n,,n!6
n用极限存在的:夹逼准则:证明数列的极限lim,0( nn,,2
111,, 求数列的极限lim(?)(222n,,n,n,n,n12
32nnsin!求数列的极限lim( n,,n,1
2x,,11123,eln() lim求数列的极限,,?,(求极限(lim ,,2223xn,,x,,,(n1)(n2)(2n),,ln()32,e,,
63ln()xx,,57xxxx,,,求极限(lim 求极限(lim 2x,,x,,,ln()xx,,34x
,,xx,,,当0,,2设,fxxgx()sin(),,2,, ,xx,,,当0,2,
讨论及(lim()lim()gxfgx,,xx,,00
设lim,(x),u,limf(u),f(u) , 证明:limf,,,(x),f(u)。 000x,xu,ux,x000
,无限循环小数0.9的值
(A)不确定
(B)小于1
(C)等于1
mn(D)无限接近1xx,求极限 、为正整数(lim() mnmnx,1 答( )2xx,,
,,若数列a适合n
a,a,r(a,a)11n,nnn,
(0,r,1)
a,ra21求证:a(lim,nn,,1,r
nxan,!n,1xan设, 其中,0是常数,为正整数 , 求极限lim nnn,,xnn
2,n求数列的极限(lim(sec) n,,n
设时,与是等价无穷小xxxx,,,()()0
且lim()()xfxA,, ,xx,0
证明:lim()()xfxA,,,xx,0
设,且,lim()fxAA,,00xx,
试证明必有的某个去心邻域存在,使得x 0
1在该邻域内有界.fx()
下述结论:
:若当时,与是等价无穷小,xxxx,,,()()0 则当时,与也xxxx,,,ln()ln()11,,,,,,0
是等价无穷小:是否正确,为什么,
xxarctan(1,),arctan(1,)应用等阶无穷小性质,求极限( limx,0x11
231416,,,xx()()1513,,,xx 求极限(lim求极限(lim 2x,0x,0xxx,211
n311,,ax522,,,xx()() 求极限 为自然数((lim()na,0求极限(limx,0x,3xx,3
设当时,与是等价无穷小,xxxx,,,()()0
fx()fxx()(),,1 且,,lim,,alim,Axxxx,,00()xgx(),
fxx()(),,证明:(lim,Axx,0gx()
设当时,,是无穷小xxxx,,,()()0
0且()()xx,, ,,
,,()()xx证明:(eexx,,~()(),,
若当时,与是等价无穷小,xxxx,,,()()01
()()xx是比高阶的无穷小(,, 则当时,与是xxxxxx,,,()()()(),,,,01
否也是等价无穷小,为什么,
设当时,、是无穷小,xxxx,,,()()0
且()().xx,,0,, 11证明:ln()ln(),,,xx,,,,,,
与是等价无穷小(()()xx,,,
设当时,是比高阶的无穷小(xxfxgx,()()0 证明:当时,与是等价无穷小(xxfxgxgx,,()()()0
若时,()与()是等价无穷小,x,x,x,x01
()与()是同阶无穷小,但不是等价,x,x
无穷小。试判定: ()()与()()也是等价无穷小,x,,x,x,,x1
吗,为什么,
确定及,使当时,Anx,0
22n()ln()()fxxxgxAx,,,,1与, 是等价无穷小(
n设f(x),sinx,2sin3x,sin5x, g(x),Ax, 求A及n,使当x,0时,f(x)~g(x)(
222()()axaxa,,设,为常数fxeeea()(),,,2
ngxAx(), 求及,使当时,Anxfxgx,0()~().
设,fxxxx(),,,,,221
A ,gx(), kx
确定及,使当时,(kAxfxgx,,,()~()
3设,,()xxx,,,32
n ,()()xcx,,1 ,
确定及,使当时,cnxxx,1()~(),,
11证明不等式:(其中为正整数ln()()1,,n nn
1xx1ab,xbxx00求极限,,lim()() ab,, 求极限,,为正的常数lim()()axeab,x,0x,02
nlnlnxx,0x,1求极限 lim()0 x,求极限,为任意实数(lim() n0xx,0x,1xx,x,103xxaa,1a,a求极限 ,(lim() aa,,01求极限lim,(a,0,a,1) x,0,xaxx,a3xx,5xtanxxee,ee,,2e,1求极限(lim 求极限(lim 求极限(lim 2,0x,0x,0xsinxxx1x1,xa2x求极限 ,且,,lim()() ababab,,,,,0011xx,01,xb11ln(sectan)xx,2xx,1求极限(lim 求极限 ,(lim()()xaaaa,,,01x,0x,,,sinx
bax求极限 ,为常数,且limln()ln()().110,,,eaba x,,,x
ln(x,x),ln(x,x),2lnx000 求极限lim (x,0)(02x,0x
1,,cosxx,x,求极限(limcos ,,求极限 ,(lim()(),,, kkzx,,,x,,x,2cos
2121xx,,21x,xx3x求极限(lim() 求极限(lim() 求极限lim()12,x2x,,x,,x,021xx,,21x,cotx2tanx1,,, 求极限lim(sin)x求极限(limtan(),xx , 求极限(lim(sincos)xx,,,,x,0x4,,x,02
112xx 求极限(lim(cos)x求极限(lim()1,,xxx,,0x,0
lncosx求极限(lim 求极限lim()ln()()ln()lnxxxxxxx,,,,,,22211,,2x,0x,,,x2x,1 求极限(lim 求极限lim,,ln(1,x),ln(x,1)x(x,,1lnxx,,,
1n1n 求数列的极限(limln()lnnnn,,1求数列的极限lim().,e,,nn,,n,,
abnn 求数列的极限limn(e,e),其中a,b为正整数(,,n
11,,2 求数列的极限limnln(a,),ln(a,),2lna ; 其中a,0是常数,,n,,nn,,
21n,1nn 求数列的极限(lim() 求数列的极限,其中(lim()naa,,10n,,n,1n,,
11(2,)(2,),,22nn求数列的极限limne,e,2e( ,,n,,,,
nn21n,ab,nn求数列的极限(lim() 求数列的极限,其中,(lim() ab,,00n,,n,,21n,2n(n,1)2,,3n2,22求数列的极限lim ,,2 计算极限:(limsin()na,,,n,,3n4,,,n,,
11设,,,则有fxx()sinsinlim()lim(),,,,xfxafxbxx,,,0xx
()()AabBab,,,,1112, ,
()()CabDab,,,,, ,2122
答( )
xxnx222eeeln()ln()1,,,?11,,,,,xxxx计算极限limln 计算极限lim x,0x,0xnseccosxx,
211,,,xxtanmx计算极限lim 求极限 ,为非零常数lim() mnx,0x,011,,xsinnx
2xaxa,,,1,xcos计算极限 lim()0 a, 计算极限(lim22xa,,0x,0cos1,xxa,
ln()ln()lnaxaxa,,,,2111计算极限在 lim()计算极限lim(,) a,02x,0x,0xxxsintanx
42sinx()ex,,,11计算极限lim 2,0x(cos)ln()11,,xx
sinxlim,x,,x
()()()()10 不存在但不是无穷大,ABCD 答( )
1limsinx之值x,,x
,,,,10 不存在但不是无穷大()()()()ABCD 答( )
AxBxtan(cos),,1已知 其中、、、是非常数lim(),10ABCD2,xx,0ln()()CxDe121,,,
则它们之间的关系为
()()()() ABDBBDCACCAC,,,,,,2222
答( )
n242 设x,1计算极限lim(1,x)(1,x)(1,x)?(1,x)n,,
x2n,121x2设及存在,试证明:(limlimx,,,01aa 求lim(sincos), nn,,,,nxx,,xxn
3232()xaxa,,,1xxx,,,332计算极限 lim() 计算极限lim a,0222xa,x,2xa,xx,,2
xxxcosee,xxx,,计算极限lim limlim(coscoscos)计算极限?22n,0x,,,,,x0n1ln()xx,,2,,22
an,1,,设有数列a满足a,0及lim,r (0,r,1),试证明lima,0( nnnn,,n,,an
n,,设有数列a满足a,0且lima,r, (0,r,1),试按极限定义证明:nnn,,n
lima,0( nn,,
设limf(x),A (A,0),试用",,,"语言证明limf(x),A( x,xx,x00
12 试问:当时,,是不是无穷小,xxx,,0,()sinx
设limf(x),A,limg(x),B,且A,B,试证明:必存在x的某去心邻域,使得0x,xx,x00
在该邻域为f(x),g(x)(
3ln()12,,x11计算极限(lim 设,试研究极限fxx()sinlim, 23x,2x,0arcsin()344xx,,xfx()
2n,,,,,nn1(1),x设数列的通项为,nn
,,nx则当时,是n
A()无穷大量
B()无穷小量 C()有界变量,但不是无穷小
D()无界变量,但不是无穷大
答( )
以下极限式正确的是
11xx,1()lim()()lim()AeB e1,,,,1x,,,,00xxx 11x,1,x()lim()()lim()C1,,,,eD 10x,,x,,xx 答( )
设, ,,,求(xxxnx,,,,10612()lim?11nn,nn,,
ax,e,1,当x,0,设f(x),,且limf(x),Ax,x,0,b, 当x,0,
则a,b,A之间的关系为
(A)a,b可取任意实数,A,1
(B)a,b可取任意实数,A,b
(C)a,b可取任意实数,A,a
(D)a可取任意实数且A,b,a
答:( )
ln(1,ax),,当x,0,设f(x)d,,且limf(x),A,x,x,0,b , 当x,0,
则a,b,A之间的关系为
(A)a,b可取任意实数,A,a
(B)a,b可取任意实数,A,b
(C)a可取任意实数且a,b,A
(D)a,b可取任意实数,而A仅取A,lna
答:( )
,ax1cos,x,,当0,2fx,fx,A设(),且lim()x,x,0,bx,, 当0,
abA则,,间正确的关系是
aAabA,(),可取任意实数2
2aBabA, (),可取任意实数2
aCab,A,()可取任意实数2
2aDab,A,()可取任意实数2
答( )
设有,,且在的某去心邻域lim()lim(),,xafAx,,0xxua,,0
内复合函数有意义。试判定是否fxfxA()lim(),,,,,,,xx,0
成立。若判定成立请给出证明;若判定不成立,请举出例子,并指明应如何加强已知条件可使极限式成立。
2,2xxb,,1,当x,,设fx()lim(), 适合fxA,x,1,x,1,ax, 当,1,
则以下结果正确的是
()AabA仅当,,,,,,434
()BaAb仅当,,可取任意实数,,44
()CbAa,,,34,,可取任意实数
()DabA,,都可能取任意实数
答( )
,11,,bx0 当x,,3设fx()lim(), 且,则fx,x,x,0,ax 当,0,
()Aba,,33, ()Bba,,63,
3()Cba,,可取任意实数
6()Dba,,可取任意实数
答( )
12cosx3设,(x),(1,ax),1,,(x),e,e,且当x,0时,(x)~,(x),试求a值。
xx,ee,2x,2ax求(lim 设lim(),8,则a,____________( xx,x,,,,xx,a34ee,2xsin lim(1,3x),____________(x,0
2当时,在下列无穷小中与不等价的是xx,0
2()cos()lnAxBx121,, xx,22()()CxxDee112,,,,,
答( )
当时,下列无穷小量中,最高阶的无穷小是x,0
22()ln()()AxxBx,,,,111 xx,()tansin()CxxDee,,, 2 答( )
2211,,xx3,54计算极限lim 2lim,sin,_____________________ xx,0x,,xxcos5,3ex,
nn,12x,x,,x,x,n?lim计算极限 x,1x,1
3n(x,1)(x,1)?(x,1),计算极限 lim xn,1 计算极限 lim(cosx)(x,1(x,1)0x,,
11讨论极限的存在性。limarctan 研究极限的存在性。limarccot x,1x,0xx,1
2xx,,23 研究极限(limx,,x,1
当x,,0时,下列变量中,为无穷大的是
sinx11(A) (B)lnx (C)arctan (D)arccot xxx
答( )
1。 lim,________________x,1xln,1
设a,0,且lima,0,试判定下述结论"存在一正整数N,使当n,N时,恒有nn,,n
a,a"是否成立, n,1n
若试讨论是否存在,limlimaAa,nnn,,,,n
,,设有数列 a 满足lim(a,a),0,试判定能否由此得出极限lima存在的,nn1nn,,,,nn
结论。
an,1,,设有数列a满足a,0;,r,0,r,1,试证明lima,0 nnnn,,an
f(x) 设lim存在,limg(x)存在,则limf(x)是否必存在,x,xx,xx,xg(x)000
f(x) 若limf(x),0,lim,A,0,则是否必有limg(x),0(x,xx,xx,xg(x)000
当x,,0时,下列变量中为无穷小量的是
11A()sin22xx
(B)ln(x,1) 1C()lnx
1xD,x,()(1)1
答( )
gx() xxfxgxAA设,时,(),,,(),(是常数),试证明lim,0(0x,xfx()0
fx()若,且在的某去心邻域内,,lim()()limgxxgx,,,00A0xxxx,,00gx()
则必等于,为什么,lim()fx0xx,0
若,不存在,则lim()lim()lim()()fxAgxfxgx,,xxxxxx,,,000
是否必不存在,若肯定不存在,请予证明,若不能 肯定,请举例说明,并指出为何加强假设条件,使
可肯定的极限时必不存在。fxgxxx()()(),,0
若limf(x),,,limg(x),A,试判定limf(x),g(x)是否为无穷大, x,xx,xx,x000
,,设x,x,f(x),,,g(x),A,试证明limf(x),g(x),,( 0x,x0
设当x,x时,f(x),,,g(x),A(A,0),试证明limf(x)g(x),,( 0x,x0
x,1设,,则当时,,,ln,,,,arcctgxxx
()~A ,,
()B与是同阶无穷小,但不是等价无穷小 ,,
()C是比高阶的无穷小,,
()D与不全是无穷小,,
答:( )
11fx()sin(),,,,,, 0xxx
()Ax当时为无穷小,,,
()Bx当时为无穷大,,0 ()()()Cxfx当,时有界,,,0
()()Dxfx当时不是无穷大,但无界(,,0
答( )
2x若,当时为无穷小,则fx(),,,,,axbx1x,
1111()()AabBab,,,,,, ,
1111()()CabDab,,,,,,,, ,
答( )
1x,12n,x32,,,求lim()? 求lim() 222n,,x,,,x6nnnn,,1,,2nnn,,n,2nlim(),____ n,,n,1
n,121
nnn?limeeee,,,n,,
21 ()()()()ABeCeDe 答( )
lim(1,2,?,n,1,2,?,(n,1)),____(n,,
2limxcos2x,,0x
(A)等于0 ; (B)等于2 ;
CD()为无穷大 ; ()不存在,但不是无穷大 . 答( )
1,设f(x),sin,试判断:xx
(1)f(x)在(0,1),内是否有界 ; (2)当x,,0时,f(x)是否成为无穷大 .
设,试判断:fxxx()cos,
()()10fx在,上是否有界,, ,,
()()2当时,是否成为无穷大xfx,,,
1试证明不存在。limcos x,0x
若在x的某去心邻域内f(x),,(x),且lim,(x),0,试证明limf(x),0 0x,xx,x00
若在x的某去心邻域内f(x),g(x),且limf(x),A,limg(x),B ; 试证明A,B( 0x,xx,x00
1sinxlim之值x,01
x AB()等于1 ; ()等于0 ;
CD()为无穷大 ; ()不存在,但不是无穷大 . 答( )
1,x3设(),()33,则当1时( ),x,,x,,xx,1,x
()()与()是同阶无穷小,但不是等价无穷小 ;A,x,x
()()与()是等价无穷小 ;B,x,x ()()是比()高阶的无穷小 ;C,x,x
()()是比()高阶的无穷小 .D,x,x
答( )
324x,ax,x,设lim,A,则必有1x,x,1
(A)a,2,A,5 ; (B)a,4,A,,10 ; Ca,A,,Da,,A,()4,6 ; ()4,10 .
答( )
121x,x,1当1时,()的极限x,fx,e1x,
()等于2 ; ()等于0 ;AB (C)为, ; (D)不存在但不是无穷大 .
答( )
322设当x,0,,(x),(1,ax),1和,(x),1,cosx满足,(x)~,(x)(试确定a的值。
2x32,求,使ablim()axb ,,,1x,,x,1
2 设lim(3x,4x,7,ax,b),0 , 试确定a,b之值。x,,,
设x,1,x,2x,3(n,1,2,?),求limx1n,1nn,,n
设, ,,,求(xxxnx,,,,42312()lim??11nn,nn,,,
n1,,,nn,1计算数列极限limtan(), n计算极限lim(arctan,arctan) ,,n,,4nn,,nn,1,,
3333k设当x,0,,(x),1,x,1,x~Ax,试确定A及k(
,()x设,求与使,()limxxxxAK,,,,,,,2210AA() kx,,,xbxx极限lim(1,) (a,0,b,0)的值为 ,0xa
bbbeaA( B Ce( D()1()ln()() aa
答( )
2x1,,设 ,试确定,之值。lim()aab0 22x,02,,(cos)axbx
2 设,试确定,之值。lim()312xaxbxab,,,,x,,,
32xaxxb,,,3设,试确定,之值。lim ,ab2x,11x,
1xsinxcos2x,,lim计算极限 计算极限lim(x,x,x,x)x,0x,,,xtanx
tansincos44,,,xx22,ax计算极限lim研究极限的存在性。lim() a,0tansinxx,0x,0xxee,2 ,,设x,(0,2),x,2x,x((n,1,2,??),试证数列x收敛,并求极限limx(n,nnnn11,,n
2 设,,,,试研究极限(xxxxnx,,,,0212()lim??11nnn,nn,,
2 设x,2,x,2x,x(n,1,2,??),试研究极限limx(n,nnn11,,n
a,bnn设a,b是两个函数,令a,ab,b,, (n,1,2,?)试证明:,,nnnn11112
lima存在,limb存在,且lima,limbnnnn,,,,,,,nnbnn
cosx,,ee,计算极限 limx,x,x,x,xx ,, 计算极限lim2x,,,,x0x,,
21xlim(1)计算极限,, 2,,xxx
若limxy,0,且x,0,y,0,则能否得出:limx,0及limy,0至少有一nnnnnnn,,n,,n,,
式成立:的结论。
设数列,都是无界数列,,xyz,xy,,,,nnnnn
试判定:是否也必是无界数列。,,z n
如肯定结论请给出证明,如否定结论则需举出反例。
31,,计算极限limsinln()sinln()x1,,,1 ,,x,,xx,,
12x极限lim(cos)x,x,0
1,201ABCDe(; ( (; (( 答( )
xx,ee,极限的值为( )lim2x,01()xx,
0123(; (; (; ((ABCD 答( )
1cos3,x极限lim的值为( )x,0sin3xx
123A(0; B(; C(; D(( 632
答( )
下列极限中不正确的是
,cosxtan3x3,2A(; (;lim,Blim,,xx,,,01sin,2x2x12
2,1arctanxxlim,,20limC(;((Dxx,,,1sin(),1xx
答( )
2211,,,,,xxxxln()ln()极限lim,2x,0x
ABCD(; (; (; ((0123 答( )
1
x极限lim(cos)x,x,0
11,2201ABeCDe(; (; (; (( 答( )
当x,0时,与x为等价无穷小量的是
A(sin2x; B(ln(1,x); C(1,x,1,x; D(x(x,sinx)( 答( )
1,x当x,1时,无穷小量是无穷小量x,1的1,2x
A(等价无穷小量;B(同阶但非等价无穷小量; C(高阶无穷小量;D(低阶无穷小量(
答( )
n当x,0时,无穷小量2sinx,sin2x与mx等价,其中m,n为常数,则数组
(m,n)中m,n的值为
A((2,3); B((3,2); C((1,3); D((3,1)( 答( )
1x已知,则的值为lim()1,,kxekx,0
1ABCD(; (; (; ((11,2 2
答( )
x12极限的值为lim()1,x,,2x
1,,144AeBeCeDe(; (; (; ( 答( )
下列等式成立的是
212222xxA(; (;eBelim()lim()1,,,,1x,,,,xxx 112212x,x,C(;((lim()lim()1,,,,eD1ex,,x,,xx
答( )
1x极限lim(12x),,0x,
122,(; (; (; ((AeBCeDe e
答( )
x,1x,4极限的值为( )lim()x,,x,1
,,2244(; (; (; ((AeBeCeDe 答( )
21x,21x,,,极限的值是lim,,x,,,,21x,
1,,22(; (; (; ((ABeCeDe1 答( )
下列极限中存在的是
2,1111xlimlimlimsinlim A(; (;(; (BCxD1xxx,,,,,,xx00xxx21,1,e
答( )
tansinxx,极限的值为lim3x,0x
110(;( ( ((ABCD, 2b
答( )
xsin极限lim,x,,,,x
101(; (; (; ((,,ABCD 答( )
ax,1cos已知,则的值为lim,ax,0sinxx2
ABCD(; (; (; ((0121, 答( )
sinkx,,3已知,则的值为limkx,0,2()xx
3,,,366(; (; (; (( ABCD2
答( )
21x,设lim()0,则常数,的值所组成的数组(,)为,ax,b,ababx,,1x,
A((1,0); B((0,1); C((1,1); D((1,,1)(
答( )
24x,3设f(x),,ax,b,若limf(x),0,则x,,x,1
a,b的值,用数组(a,b)可表示为
A,B,CD,,((4,4); ((4,4); ((4,4); ((4,4) 答( )
2xx,,68极限的值为lim2x,2xx,,812
1(; (; (; ((ABCD012 2
答( )
下列极限计算正确的是
2nxxx,sinA(; (;limlimB,11,2nn,,,,,xxxsin,x1, xx,sin1n2C(; ((Delim,,,01lim()3xn,,,0n2x
答( )
32xx极限的值为lim(),2x,,x,1,1x
011,,ABCD(; (; (; (( 答( )
2数列极限的值为lim()nnn,,n,,
1ABCD(; (; (; (不存在(01 2
答( )
2xxc,,3已知,则的值为lim,,1Cx,1x,1
ABCD(; (; (; ((,1123 答( )
26x,ax,已知lim5,则的值为,a1x,1,x
A(7; B(,7 C(2; D(,2( 答( )
x,e,2, x,0
,设函数f(x),1, x,0,则limf(x),,x,0,x,cosx,x,0,
,110A(; B(; C(; D(不存在( 答( )
1,cosx,,x,0,x,设(),,则fx ,,1x,x,0,1x,1,e,
,A(limf(x)0;x,0
,B(limf(x)limf(x);,,x,0x,0
C(limf(x)存在,limf(x)不存在; ,,x,0x,0
Dfxfx(lim()不存在,lim()存在(,,x,0x,0
答( )
tankx,,,x0,x,设f(x),且limf(x)存在,则k的值为 ,x,0,,,x3,x0,
A(1; B(2; C(3; D(4( 答( )
下列极限中,不正确的是
1x(lim(1)4;(lim0;Ax,,Be,,,x,3x,0
1sin(1)x,1x(lim(),0;(lim,0(CD x,0x,12x
答( )
f(x)g(x)若lim,0,lim,c,0(k,0)( kk,1x,0x,0xx
则当x,0,无穷小f(x)与g(x)的关系是
A(f(x)为g(x)的高阶无穷小;
B(g(x)为f(x)的高阶无穷小; C(f(x)为g(x)的同阶无穷小;
D(f(x)与g(x)比较无肯定结论(
答( )
2当x,0时,2sinx(1,cosx)与x比较是( ) A(冈阶但不等价无穷小; B(等价无穷小;C(高阶无穷小; D(低阶无穷小(
答( )
3当x,0时,sinx(1,cosx)是x的
A(冈阶无穷小,但不是等价无穷小; B(等价无穷小;C(高阶无穷小; D(低阶无穷小( 答( )设有两命
题
快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题
:
,,,,命题"a",若数列x单调且有下界,则x必收敛;nn
,,,,,,,,,,,,命题"b",若数列x、y、z满足条件:yxz,且y,z都有收敛,则nnnnnnnn
,, 数列x必收敛n
则
A("a"、"b"都正确; B("a"正确,"b"不正确;
C("a"不正确,"b"正确; D("a","b"都不正确(
答( )设有两命题:
命题甲:若lim()、lim()都不存在,则lim()()必不存在;fxgxfxgx,,,x,xx,xx,x000
命题乙:若lim()存在,而lim()不存在,则lim()()必不存在。fxgxfxgx,x,xx,xx,x000则 (甲、乙都不成立; (甲成立,乙不成立;AB
C(甲不成立,乙成立; D(甲、乙都成立。
答( )设有两命题:
()fx命题"":若lim()0,lim()存在,且()0, 则lim0;afxgxgx,,,0x,xx,xx,x()gx000命题"":若lim()存在,lim()不存在。则lim(()())必不存在。bfxgxfxgx,x,xx,xx,x000
则
("",""都正确; (""正确,""不正确;AabBab
(""不正确,""正确; ("",""都不正确。CabDab
答( )
若lim,f(x),,,limg(x),0,则limf(x),g(x)x,xx,x00x,x9
A(必为无穷大量 ; B(必为无穷小量 ;
C(必为非零常数 ; D(极限值不能确定 ( 答( )
,,,,设有两个数列a,b,且lim(b,a),0,则 nnnnn,,
,,,,Aab ;(,必都收敛,且极限相等nn
,,,,B(a,b必都收敛,但极限未必相等 ;nn
,,,,C(a收敛,而b发散 ; nn
,,,,Dab(和可能都发散,也可能都收敛(nn
答( )
下列叙述不正确的是
A(无穷小量与无穷大量的商为无穷小量;
B(无穷小量与有界量的积是无穷小量;
C(无穷大量与有界量的积是无穷大量; D(无穷大量与无穷大量的积是无穷大量。
答( )
下列叙述不正确的是
A(无穷大量的倒数是无穷小量;
B(无穷小量的倒数是无穷大量;
C(无穷小量与有界量的乘积是无穷小量; D(无穷大量与无穷大量的乘积是无穷大量。
答( )若limf(x),,,limg(x),,,则下式中必定成立的是 x,xx,x00
(lim()() ; (lim()()0 ;A,,,,fx,gx,,Bfx,gx,x,xx,x00
()fxC(lim,c,0 ; D(limkf(x),,,(k,0) ( x,xx,xg(x)00
答( )
1设函数f(x),xcos,则当x,,时,f(x)是 x
A(有界变量; B(无界,但非无穷大量;C(无穷小量; D(无穷大量( 答( )若limf(x),A(A为常数),则当x,x时,函数f(x),A是 0x,x0
A(无穷大量 ; B(无界,但非无穷大量 ;C(无穷小量 ; D(有界,而未必为无穷小量 (
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1设函数f(x),xsin,则当x,0时,f(x)为 x
A(无界变量; B(无穷大量;C(有界,但非无穷小量; D(无穷小量( 答( )f(x)在点x处有定义是极限limf(x)存在的 0x,x0
A(必要条件; B(充分条件;
C(充分必要条件; D(既非必要又非充分条件( 答( )
an,1,,设正项数列满足lim,0,则a nn,,an
,,,A(lima0; B(limaC0;nnn,,n,,
,,Clima; Da(不存在 (的收放性不能确定( nnn,,
答( )若lima,A(A,0),则当n充分大时,必有 n,,n
A(aA; B(aA;,,nn
AAC(a; D(a(,, nn22
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,,数列a无界是数列发散的 n
A(必要条件; B(充分条件;C(充分必要条件; D(既非充分又非必要条件( 答( )
下列叙述正确的是
A(有界数列一定有极限;
B(无界数列一定是无穷大量;
C(无穷大数列必为无界数列; D(无界数列未必发散
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