高三数学数列极限与函数极限例题解析 人教版[doc]
高三数学数列极限与函数极限例题解析一. 本周教学内容
数列极限与函数极限 二. 重点、难点
1. 数列极限的几个重要公式
若 lima,alimb,bnnn,,n,,
则
(1) lim(a,b),(a,b)nnn,,
(2) lim(a,b),a,bnnn,,
aanlim,(b,0)(3) n,,bbn
2. 数列极限的几个重要极限
(1)limc,c n,,
(2)limc,a,c,lima nnn,,n,,
1(3) (k,0)lim,0kn,,n
klimq,0(4) (q,1)n,,
3. 函数极限
(1)limf(x),a,limf(x),limf(x),a x,,x,,,x,,,
limf(x),a,limf(x),limf(x),a(2) x,x,,0xxxx,,00
0(3)f(x)为型需约分,再求极限。 0
4. 连续
x,xxy,f(x)在处连续(在左右有定义),limf(x),f(x)000n,x0
【典型例题】
nlimq,0[例1] 求证 (0,q,1)n,,
证:任意小正数 ,,0
nnq,0,,q,,解不等式 nlgq,lg,
,lg,? n lgq
,lgN,[]令([…]为取整函数) lgq
nnq,0,,故当时总有 ? limq,1 (、N
证明
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),N,N,1n,,
[例2] 下列数列极限 limann,,
3100n,na,(1) n42n,2
n4n,2,1a,(2) nnn,3,1
nn(,2),3(3) ,ann,1n,1(,2),3
2a,(n,3n,n)(4) n
(5) a,n,[n,1,n]n
1,2,2,3,?,n(n,1)(6)a, n3331,2,?,n
1,4,,(3n,2)?a,(7) n2n,1
111(8) a,n(1,)(1,),?(1,)n34n,2解:
lima,0(1) nn,,
211nn4,(),,()n343(2)a lim,lim,0nn,,n,,11n1,,()n3
121n(),,,1333(3) limlima,,nn,,n,,23n,1()1,,3
n333,,alim,lim,lim,,(4) n2n,,n,,n,,23nnn3,,11,,n
n1a(5)lim,lim, nn,,n,,2n,1,n
nnnn1(,1)2nnn(,1)(2,1),kk,,,11k,k,62(6)alim,lim,lim,0nnn,,n,,n,,3122k,nn(,1)1k,4
1(3n1)n,,32lima,lim,(7) n2n,,n,,2n1,
nn23,12anlim,lim,,?,lim,2(8) nn,,n,,n,,nn34,2,2
[例3] 填空
2ann(,2),4lim,2(1), 。 a,2n,,n2,7
2n,1lim(,an,b),1(2),则 , 。 a,b,n,,n,1
2(3),则 。 a,lim(2n,4n,an,3),1n,,
b2n(4),则 。 lim[2,()],2b,n,,b1,
解:
a,2(1) ,2a,62
1,a,0a,1,,(2) ,,,,a,b,1b,,2,,
a,1(3) a,4
2,4
2b1,1(4) b,(,1,)1,b3
n,1n,1a,2[例4] 求的极限 n,1n,1a,2
解:
12n,11(),21a2a(1) a,2lima,,n,,121n,1(),222aaa
a11n,1(),,1244(2) lim,,,a,2n,,a114n,1()1,24
n,1n,1223,lim,3) (a,2n,1n,1n,,522,
n,14,(,1),1(4) 无极限 lima,,2n,1n,,(,1),4
[例5] 各项均为正数的等比数列,它的每一项均等于后面所有项之和,求公比,
*解: n,N
aq1n,1a,a,a,a,?1, q,nn,1n,2n1,q1,q2
[例6] 求下列函数极限
23x,2x,5lim(1) 2x,,5x,8x,1
2lim2(sinx,cosx,x)(2) ,x,2
2x,5x,6lim(3) 2x,3x,8x,15
x,1,2lim(4) x,3x,3
22sinx,sinx,1lim(5) 22sinx,3sinx,1,x,6
13,(6) lim()3x,,1x,1x,1
解:
253,,23xx(1) lim,x,,8155,,2xx
224,,,2(2)lim2(sincos)2(10) xxx,,,,,,42,x,2
(x2)321,,lim,,,(3) x,3(x5)352,,
x1211,,(4) limlim,,22x,3x,34(x1)2x12,,,,
xsin,1lim,,3(5) xsin,1,x,6
xxx(,2)(,1),2(6)lim,lim,,1 22x,,1x,,1xxxxx(,,1)(,1),,1
[例7] 研究下列函数的极限(在处) x,0
x,2x,0
,fxx(),0,0(1) ,
,2xx1,,0,
x,1x,0,
,f(x),0x,0(2) ,
,x,1x,0,
解: (1)limf(x),1 limf(x),1 ? limf(x),1 x,0,,x,0x,0(2)limf(x),1 limf(x),,1 ? 无极限 f(x)x,0,,x0x,0,
[例8] 求、 ab
2(1) lim(a2x,x,1,bx),1x,,
2,x,1x,1,y,f(x),limf(x),b(2) 且 ,3x,1,2x,ax,1,
22x,ax,lim,b(3) x,,22x,
2axbx,,1lim,3(4) x,1x,1
解:
222222222axxbxabxaxa(2,,1),(2,),,(1) 即lim,1lim,122x,,x,,axxbxaxxbx2,,1,2,,1,
22,b,42a,b,a,,, ? ,,2a,22,a,2a,b,,
2) (limf(x),2,alimf(x),0,,x1x,1,
2,a,0a,,2,, ? ,,,b,0b,0,,
2(3)由已知为的因子。 (x,2)x,ax,2
2x,ax,2,(x,2)(x,1) ? ? a,3
(x,2)(x,1) lim,,1,bx,,2x,2
2(4)由已知为的因子。 (x,1)ax,bx,1
2ax,bx,1,(x,1)(ax,1) ?
xax(,1)(,1) ? lim,3x,1x,1
a,b,1,0a,4,, ? ,,,a,1,3b,,5,,
2,x,2x,3x,1
,yx,20,,1[例9] 求函数的连续区间 ,
,x,1x,0,
解:limf(x),2 limf(x),1 在处不连续y,f(x)x,0,,x0x,0,
limf(x),2limf(x),2 f(1),2 ,,x,1x,1
? 在区间(,,,0),[0,,,)上连续
[例10] 求a
2,x1,,1x,0,32,(1)y,f(x)在R上连续 ,x1,,1,ax,0,
,1,1,xx,0,(2)y,f(x),R上连续 ,x
,a(x,1)x,0,
62解:令 t,1,x
32t,1t,t,133limf(x),lim,lim1) ? (a,,,f(x),a2x,0t,1t,1t,1t,122
x1limf(x),lim,(2) 2,,x(11x),,x,0x,0
1 ? limf(x),af(0),aa,2,x0,
,,a[例11] 数列中,,求,a,2,2a,2,alimaa,2n2nn,1n1n,,
22解:令lima,lim(2,a)lima,t ? ? a,2,ann,1nnn,1n,,n,,n,,
2? ? (舍) t,2,tt,2t,,1
,,,另解:2cos 2cos 2cos a,a,a,12nn,1482lima,2cos0,2? nn,,
111lim(,,?,)[例12] n,,2n2nn
111f(n),(1,,?,)解:令
n2n
2122,,, k,k,1kk,kk,1,k||||
2(k,1,k)2(k,k,1)
122a,(,?,)令 nn1,2n,n,1
2,[(2,1),?,(n,1,n)]
n
2,(n,1,1)
n
2111,,,?,b()令 n,,,,n0112n1n
2,[1,(2,1),?,(n,n,1)]
n
*a,f(n),b? ? lima,limf(n),limbn,Nnnnnx,,x,,x,,
? limf(n),2x,,
,,,,,,
[例13] 无穷等比数列:,,,…求所有项和。 0.150.00150.015
,,0.151551解:a,0.15 ,,,q,111099331,100
5
5033? S,,12971,10
[例14] 无穷等比数列,各项和为9,各项平方和为27,求公比。
a,19,,9a,,1,1,q,,2,解: ,,21a1,,q,,272,,2,1,q,
nmmxnx(1,),(1,)1. 、,、,求极限 limmmn,Nn,22x,0x
f(x)f(x)fx()2. 为关于的三次四项式,且,,求lim,limf(x)f(x)xlim,,2x,2x,1x,3xx,1x,3,2
xab,,lim,13. ,求、 ab2x,1x,1
12n,,aaa4. 数列的相邻两项,是方程的两根,且a,2,x,cx,(),nnn,11n3
S,c,c,?,c,求 limSn12nnx,,
P5. 中,,,,在BC边上有个等分点,PP…,求n,ABCAB,cBC,aCA,bn12
1222 lim[AP,AP,?,AP]12nx,,n
A
CBPk
[参考答案]
12233122CmxCmxCmxCnxCnx[1,,(),()],[1,,(),?]nnnmmlim1. 2x,0x
222223CmCnxx(,),(),?nm,lim 2x,0x
12222 ,Cm,Cn,mn(n,m)nm2
2. 由已知f(x),a(x,1)(x,2)(x,3)
f(x)又 ? , lim,,a,,2a,2x,2x,2
fxfx()() lim,4lim,4x,1x,3xx,1,3
2x,a,bx,a,b3. 显然不成立 ? ? ,b,0b,022x,1(x,1)(x,a,b)
15,a,,2,,a,b,1,,16, ? ,,1,2(a,1,b),1,,b,,,4,
,,aaC,nn,1n11,4. a,2a,a,a,112n,1n,1n63aa(),,nn,1,3,
Caa,12k,22k,32k,2,, Caa3,2k2k,12k
C13512k,1,同上, C,C,12C3662k,1
135,966 limlim[()()]S,c,c,c,c,?,,n1234x,,x,,121,3
,ABP5. 中 k
kaka22222,,,, c()2ccosBAP,AB,BP,2AB,BP,cosBkkk,,n1n1
a2ac222 ,c,()k,cosB,kn,1n,1
a2accosB22222222,nc,(),(1,2,?,n),? AP,AP,?,AP12kn,1n,1
(1,2,?,n)
2an(2n,1)2,nc,,accosB,n 6(n,1)
21a2n,12222lim[AP,AP,?,AP],lim[c,,,accosB] n12x,,x,,n6n,111122222,c,a,accosB,(b,c),a326