教程說明所示

2012-10-第一節(jié)數(shù)列的極限一、數(shù)列極限的二、收斂數(shù)列的13-

2012-10- 設(shè)有半徑為r的圓，用其內(nèi)接正n邊形面積 近圓面積Sr rA

1r

sin

(n3,4,5, 近S0,N`,nN

An .13-2012-10-（數(shù)列極限的N定義）若數(shù)列xna0,N`,當(dāng)nN時(shí)恒 xna.xn的極限a, limxn a(n n幾何解釋幾何解釋a(xNa)Na直觀解釋:aa13-例1.

xn

nn

,證明數(shù)列xn

2012-10-證

xn

n(1)n 0

xn

,即1,n 因此 取N1`

則當(dāng)nNn

1

故lim

n

13-例

0.(n1)2

2012-10-證

(n1)2

(n n1n 令1,n1則對0,取N1`,nnN

xn0.

注：(1)N有關(guān)：越小，N越(2)NN

xna

P2113-例3.設(shè)0q1

2012-10-證:

qn1

q對0,不妨設(shè)0

qn1解得.故可取N[1 ln]`ln lnnN

xn

注：(1)(2)為了便于找N，可不妨設(shè)0k13-

2012-10-

n 2 2

3n2n 提示:Nnn

提示

P49Ex13-

2

2012-10-b證:（反證法）limxna,limxnb, 2且a2

N1`,當(dāng)nN1N2`,當(dāng)nN2

xnaxn

..對NmaxN1,N2`, 則當(dāng)n>N時(shí)ba(xna)(xn xnaxnb2b13-2012-10-性質(zhì)2.收斂數(shù)列一定有界證:設(shè)limxnaa a (xna)a

取1,N`, 當(dāng)nN時(shí)取Mmaxx1

x2,",

,1

a

(n1,2,")

注:(1)(2)逆否命題 13-性質(zhì)

2012-10-若limxna0(若則N`,nN

0(證 對a>0,取a,則N`,當(dāng)nN時(shí)2x 0aaaxa a0a,2注:收斂數(shù)列（幾乎所有項(xiàng)）13-2012-10- 若limxna,且N`, 當(dāng)nN時(shí)nxn0( 則a0( 推論2.收斂數(shù)列具有保序性若limxna,limynb 且N`, nNxn(yn,則a(（1易證13-2012-10-性質(zhì)4.（P20定理limxn （證略

limx2k1limx2kk k13-2練習(xí) 2

2012-10-

n

為了有利于放大，常先設(shè)n5n10n 2n5n10提示

n2n2n3n22nnn

1 (n2)3(3n22n4) 33n2 n1n

nn nnn n11"n

(n2)1n

2 nnnnnP49Ex13-xnf

yf

2012-10- 一、自變量趨于無窮大時(shí)函x,x,x二、自變量趨于有限值時(shí)函 xx,xx x 9-2012-10-定義1.

0,X0當(dāng)

X時(shí)，f(x

則稱A為f(x)當(dāng)x

limf(x)xyAA

f(x)A(xA

yf yA為曲線y=f(x)的水平漸近線9-例1.證明lim1x

2012-10-證:

1

1

由1解得

1 x0,

取X1,當(dāng)

X有10.xy注:(1y0y1的水平漸近線x

yx P269-2012-10-二、自變量趨于有限值時(shí)函數(shù)的極限（定義

設(shè)f(x)在x0的某去心鄰域0,0,當(dāng)0 x

f(xAAf(x當(dāng)xlimf(x)x

yf f(x)A(xx0

AAA

x0 9-例2.limCC,C為常數(shù)x例3.lim(2x1

2012-10-證:

f(x)

(2x1)

2x1由2x1

解得x

,故0,2取02

x

(2x1)

.P26Ex1(3)P26

9-x例4.證明:x

x

xx0xx0xxx

x

2012-10-取P24例

minx0,limsinxsinx0.x

P24例

limlog

logx0

xa0ax

注:對某些函數(shù)而言，函數(shù)值9-

2012-10-limf(x)A0,0,s.t.0x

x

f(x)

.xlimfxlimf(x)A0,0,s.t.x0xx0xlimf(x)A0,0,s.t.x0xx00f(x)A.f(x)A.0limf(x)A0,X0,s.t.xX f(x)Alimf(xlimf(x)A0,X0,s.t.xX f(x)Alimf(x)A0,X0,s.t.xXf(x)Alimxna0,N`,s.t.n xna9-左極限：limf(x0x0右極限：limf(x0x0

f(x)00f(x)00

ff

2012-10-

limf(x)Alimf(x)limf(x) x x x limf(x)Alimf(x)limf(x) 注：如：設(shè)f(x)a x1,且limf(x)存在，2x x a 9-x

x

2012-10-例5.設(shè)函數(shù)f(x

x0,討論x0fxy解 因f(0)lim(x1)f(0)lim(x1)

yx

1o

yxx顯然f(0f(0所以limf(x)不存在P26例7例9-2012-10-第三節(jié)極限的性一、極限的四則運(yùn)算法8-定理1.limf(x)A,limg(x)B,lim[f(x)g(x)]ABlimf(x)limg(x)lim[f(x)g(x)]ABlimf(x)limg(x)limf(x)Alimf(x).(Bg(x) limg(x)

2012-10-

lim[Cf(x)]Climflim[f(x)]n[limf(x)]n

C8-推論3.（數(shù)列極限的四則運(yùn)算法則

2012-10-例1.若Pn(xa0a1x"axn

則limP(x)P(x x

例

若R(x)

,limR(x)R(x

其中Q(x

x

注:Qm(x00不能直接例3.

x24x3lim(x3)(x1)limx12

x2

x3(x3)(x

x3x x3時(shí)分母不為x3時(shí)分母不為0x30例4.計(jì)算

2012-10-x11

1x 2x lim 3 x11

1x

1

(1x)(x

x

x1(1x)(1xx2 x11x

lim(x)a,x某去心鄰域內(nèi)(x)a0x0 limf(u) limf[(x)] u x注:(1)共6種極限過程都適用 8-xxx2例5.計(jì)算 解:令ux3,則limulimx3

x2xxx2u6

x3x2u61u6

x3x P29例5.limcosxlimsinxsinxcosxx x

0 sin

sin

limtanx

tan

xk2,k]x

xx0cos

cos cotxcosx,secx ,cscx sin cos sin8-例

lim

x21

2012-10-x21解x21x

x

11

（P28定理8-（8定理4）歸結(jié)原理*（海涅定理

2012-10-n設(shè)n設(shè)

x0

且

xn1,2 0limf(x) 0

limf(xn)x （limf(x)A0,0,s.t.0xx0,f(x)Ax對上述0,limxnx0N`,s.t.nN,0xn 此時(shí)f(xn

.

limf(xn)即即 8-例8*.證明limsin1不存在

2012-10-解:

x

0,則limx0,limf(x 2n2

取x

0,則limx0,limf(x 2n2

注 f(x)sin1的圖象如圖 x（在-1和1之間無限次振蕩8-2012-10-第四節(jié)無窮小與無窮一、無窮二、無窮三、無窮小的12-

2012-10-定義1

若limf(x)0,f(xxx注:6種極限過程如 lim10f(x)1為x時(shí)的無窮小x limx1)0f(x)x1為x1時(shí)的無窮小

為n

0xn n(n (n12-2012-10-定理1（無窮小與函數(shù)極限的關(guān)系limf(x)

f(x)A,limlimf(x)A0,0，f(x)，f(x)A

x

f(x)

lim012-2012-10-定理2（無窮小運(yùn)算法則yysinxox如：(1lim(xyysinxoxlimarctanx sin 注：y0是ysinxx12-2012-10-定義2.若M0,0,s.t.0 xx0 f Mf(x)為xx0時(shí)的無窮大limf(x(1)limf(xM0,limf(x)M0,

""",f(x)M""",f(x)Mf f 12-例1.

lim1

2012-10-x01x證：M0,1x

M,

x

1,

0, yxy當(dāng)0x01M.yxyxo注)lim1 lim1ox

x (2)limf(x)xx為曲線yf(x)的

12-

2012-10-例2.證明f(x)xcos 但非x時(shí)的無窮大(1)M0,x[M]1\, f(x)[M]1M(2)M0,Xx[X]X2 f(x)0M

f(x)xcos 12-

2012-10-f(x) f

f(x)為無窮小且f(x)0

f

為無窮大常用結(jié)果：a0b00,mn為非負(fù)整數(shù)axmaxm1"

a0b0 mlim

m

b

b

"

m12-三、無窮小的比較

2012-10-引例

x0時(shí)，3x,x2,x都是無窮小,x2x

0,lim

1 2

,x0

x0

x00的速度是不同的0

12-2012-10-定義3.設(shè)是自變量同一變化過程中的無窮小,若lim0,高階的無窮小,記作D(,低階limC0,k階k1的同階無窮小lim1,的等價(jià)無窮小,~.12-例1證明：x0時(shí)

1～xn1n1

2012-10-n1證:n1 x/

n1xnlimnx0 n1xn1n1xn2"

注

bn(ab)(an1an2b"12-定理4.～ ,定理5（等價(jià)無窮小代換定理）

2012-10-P34~, ~,且lim存在，limlim

limlimlimlimlimlim

P3412-2012-10-第五節(jié)極限的存在準(zhǔn)一 準(zhǔn)17-

2012-10-準(zhǔn)則 準(zhǔn)則）設(shè)數(shù)列滿足N0`,nN0ynxnznlimynlimzn n證:由條件(2)0,N1N2`,

則limxn當(dāng)nN1yn當(dāng)nN2zn

故a 故zna取NmaxN0N1,N2`,則當(dāng)nNaynxnzna xna,即證17-例1.證明：limn

"

2012-10-

n2

n2

n2n

n2

n

"n2n

n2n2n

n n

2nn2

nnP35例

lim

"

nn

n2

n2n17-2012-10-例

準(zhǔn)則證明：limnnn1時(shí)記nn1h 0),

n1nh

n(nh2

"n(n1) 0h2 0h n n 1nn1h1 n

n而lim1n

n1

17-2012-10- 如 準(zhǔn)則）設(shè)函數(shù)f1(x),f(x),f2(x)滿足0,當(dāng)x0xx0時(shí)，ynxnznlimf1(x)limf2(x) x x 則limf(x)0x0注:(1)A為(26種極限類型17-例3.求證：limsinx

2012-10-

2證:x0, 時(shí)如右圖2 SAOBS扇形AOB

oxr 即1r2sinx1r2x1r2tan即 1 1 ,0x

得cosxsinx 0x2 sin cos 2

2 2∵limcosxcos0

17-例

2012-10-limtanx

limsinx

x0 cosxlimarcsinx

lim arcsin 1

x0sin(arcsinx

x1cos

2

1

lim 2 2 x2

x2 x02 注：x0sinxtanx

arcsinxarctanx

1cosx～217-例 limtanxsinx

原式limxx

2012-10- 1

(1x2)311cos

x解:(1)原式

tanx(1cos

lim 2原式

x2/3 2

x0x/2 注:x0時(shí)，(1 1～ （等價(jià)無窮小代換17-

2012-10-x1

xn 極限值即為最小的上界（上確界

證略 極限值即為最大的下界（下確界17-例6.設(shè)

1xa,n1,2,"

2012-10-且a,x 2 x 求limxn

n：

122n

axnx

xnxna xnxna

1 a

1 1

n

2 a故極限存在！設(shè)lim 則A1Aa

2 AA

∵xn0,故limxn 17-2012-10-n例7.設(shè)ai0(i1,2,"), n12x 12

1 (1a)(1a

(1a)(1a)"(1a 證:顯然{xn單調(diào)增

(1ak)xn

(1

)"(1a

(1

)"(1a

k n n

k 1 (1

)"(1 (1a)"(1a) 1

k2

k 1 1.limx 存在(1a1)"(1an

（拆項(xiàng)相消17- 1

2012-10-例8.設(shè)xn1 (n1,2,"), n分析：(1)xn(2)xn有上界（方法

P37-P38記lim

1

e,n n11xx lim1xx117-2012-10-如:（單調(diào)有界準(zhǔn)則）設(shè)函數(shù)f(x)X0當(dāng)xXf(x單調(diào)增且有上界，則limf(x)存在.注:函數(shù)的其它單調(diào)有界準(zhǔn)則較復(fù)雜甚至未必成立x 1x如：f(x) x 0xf(xlimf(x)不存在

17-2012-10-例

1 lim

nn1 ln(1

n 1 xlimex

x) lne 令uex

1

u0ln(1注:x0時(shí)，ln(1x～xex1～x17-例 1

1x

2012-10-lim

lim

x x1

x

x 1limxx1

1

lim[1(x lim x x

sinx x1lim

12

1

sin2

x

x

x

x 17-2012-10- lim12xsinx 12xsinx解:原式

12x2

e6u(x)v(x)ev(x)lnu(x),u(x) ln(12

lim32lim12xsinx limesinx

ex0 e617-

2012-10-： 代表相同：表達(dá)1型未定式x00sinxx00sinx tanx～x arcsinx～x arctanx～x ex1～xe1ln(1x)～x )1cosx～x2/1 ～2/n1x1～x/n1 1～/ lim(1 17-2012-10-練習(xí)

limsinx limxsin1 limxsin1

2lim2n1lim

2n1

n

2n1 17-2012-10-節(jié)連二、函數(shù)的間三、初等函數(shù)的四、閉區(qū)間上連續(xù)函19-

2012-10-

設(shè)yf(x在x0的某鄰域 limf(x)f(x),f(xx點(diǎn) x注1：f(x)x0;

xlimf(x)limf(x)f(x0x19-注2limf(x)f(x0x

2012-10-0,0,當(dāng)x時(shí)，f(xf(x0limlimy

xx0

yyf 函數(shù)增量：yf(xf(x0f(x)f(x)f(x000f(x)f(x)f(x0000左連 右連19-2012-10-為該區(qū)間上的連續(xù)函數(shù).f(xf(a)flimf(x)f(x),x(a,b);

（右連續(xù)x

（內(nèi)連續(xù) f(b)f

（左連續(xù)f(x為閉區(qū)間[ab上的連續(xù)函數(shù)f(x)19-

2012-10-ysin ycosx,aylogx(a0,aa

（P24例3P29例5（P24例 P(x)aax"a （多項(xiàng)式函數(shù)P28 R(x)P(x)Q(x)

（有理函 P29例2 例1.f(x)xsinx

x0,x=0則a0x19-

2012-10- 若f(x)在點(diǎn)x0不連續(xù)，即下列情形之一發(fā)生，則稱點(diǎn)x0為f(x)的間斷點(diǎn). f(x)x0無定義

limf(x不存在limf(x)f(x0

limf(x)f(x0x19-

2012-10-

f

及f(x0)均存在若 若 f(x0)f(x0稱x0為 若f )f ),稱x0為跳躍間斷點(diǎn) 第二類間斷點(diǎn):f(x0 )或f(x0 )中至少一個(gè)不存在，若其中有一個(gè)為,稱x0為 稱x0為振蕩間斷點(diǎn)若19-f(x)

x2x

2012-10-y∵limf(x)x1為其可去間斷點(diǎn)

y1oxx1, y1oxf(x) xx x∵f(0)11f(0x0為其跳躍間斷點(diǎn)19-2012-10-f(x)tan∵limf(x)2x為其無窮間斷點(diǎn)2

yo 2ysinx

ysin1xx0為其振蕩間斷點(diǎn) 19-例2.討論f(x)

x2x23x

2012-10-答案：x1是第一類（可去）x2是第二類（無窮）間斷點(diǎn)例3.討論f(x)

1e1x顯然，間斷點(diǎn)為x0,xx∵limf(x)

x0為無窮間斷點(diǎn)limf(x)0,limf(x)1.x1為跳躍間斷點(diǎn) 19-2012-10-基本初等函數(shù)初等函數(shù)連續(xù)性四則運(yùn)算定理1（連續(xù)函數(shù)的運(yùn)算性質(zhì)

（P43定理（P44定理單調(diào)連續(xù)函數(shù)的反函數(shù)仍為連續(xù)函數(shù).（P44定理19-

2012-10- xelnx(x（四則+復(fù)合結(jié)論：初等函數(shù)在定義區(qū)間內(nèi)cosxcosx19-2012-10-例4.f(x),g(x均在[ab(x)maxf(x), (x)minf(x),在[ab]上也連續(xù)證∵(x)1f(xg(xf(xg(x)22(x)1f(x)g(x)f(x)g(x)2(x(x)在[ab]上也連續(xù)注 19- x2 x

(x)

x

2012-10-例5.設(shè)f

2

x

x

x討論復(fù)合函數(shù)f[(x)]解：f[(x)]

x2 x2

xf[(x)]x≠1時(shí)連續(xù)limf[(x)]limx2 limf[(x)]lim(2x) f[(x)]x=1不連續(xù)19-2012-10-f(xCab1,2abf(1) faxf(2)maxfax推論.

yyfoa 2 b該區(qū)間上有界.（有界定理19-設(shè)f(xC[ab],mminf(x),a

2012-10-yyfMMmaxf(x),則對介于m與M aC,[a 使f(設(shè)f(xC[ab],