月6日精講2-高等數(shù)學(xué)-1262高數(shù)

?主講老師：約占約占約占約占約占?（?）函數(shù)極限連續(xù)主講老師 （自變量 唯 （因變量或函數(shù)值yf(x)0

分段繪制⑴分母偶次被開?數(shù)⑶對數(shù)⑷反正、余弦正切函數(shù)的自變量不能等于π2+kπ，余切函數(shù)的自變量不等于kπ；?個式?每?部分的交集并集⑴分母偶次被開?數(shù)⑶對數(shù)反正、余弦正切函數(shù)的自變量不能等于π2+kπ，余切函數(shù)的自變量不等于kπ；?個式?每?部分的交集并集

y x yx1

x

x xxx x xx x x分部分層(2004)yx1x2(2007)y1x2(2004)yxx 1x21x2(2007)y1x21x1x20,下列函數(shù)為奇函數(shù)的是 x2exex設(shè)函數(shù)的定義域為2則函數(shù)y1[f(x)f(x)]在其定義域上是 2

下列函數(shù)為奇函數(shù)的是 x2exex(2則函數(shù)y1[f(x)f(x)]在其定義域上是 2

設(shè)函數(shù)f(x)的定義域是全體實數(shù)則函 f(x)f(x)是

(3)設(shè)函數(shù)f(x)的定義域是全體實數(shù)則函 f(x)f(x)是 F(x)f(x)f(x)F(x)f(x)f((x

f(x)f(x)f(x)f(x)F(x F(x)是偶函

f(x)x22xf(x1)x22

則fx1)則fx)x已 f(x1)x22x f(x) f(1) f(2x

f(x)x22xf(x1)x22

則fx1)x1)22(x1)x24x則fx)設(shè)x1 則xt f(t)(t1)22(t1)t2 f(x)x2x f(x1)x22x f(x) f(1) f(2x x1t xt f(t)(t1)22(t1)5t2即f(xx26

f(1)(1)

61x2

f(2)226對數(shù)函數(shù)g三角函數(shù)：y=sinx反三角函數(shù)：y=arcsinx等（分段函數(shù)?般不是初等函數(shù)⑵收斂數(shù)列?定是有界數(shù)列；有界數(shù)列不?收斂發(fā)散數(shù)列不?定?界；?界數(shù)列?定發(fā)散⑸改變數(shù)列的有限項⑴唯?⑵⑶ ⑸⑴唯?⑵⑶ ⑸

anbncn,liman nlimbn

1(annnnlim10(anlim10(anlimqn0(q 1n n求極限

求極限

3n4(31)2(2

3 12 n4 n42n 求極限lim (2)求極限lim(2)求極限lim n 32n

n 32n3 3

2233 3 22 nlim3lim3lim32 3 0103 求極限limn2n n2n2n2

n2

n2n2n2n n2n2n2n n211n11n1 2 (4)求極限limnln(21ln (4)求極限

n(ln(21)ln2lim n limnln(11n 2nlimln(11n limln[(1 )2nn 2n1lne2考點33n3n1 3n1nnnn4n4n2n

24n3n12n4n13n2n考點33n3n1 3n1n

n3n31 3 11n1n4n2n4n2n

3 n1 3

14n2n44n2n41n

24n3n1

23(3)n

1

4n13n

n

(2) )n( )n( xx 函數(shù)某點極限存在的充要條件函數(shù)在該點的左右極限存在并且相等 xx 0

f(x)g(x) f(x) f(x)g(x) f(x) g(x

f(x)

f(xlim

xx0g(x

g(x?窮?量與? 2常數(shù)、有界變量、有極限的變量與?窮小的

lim12...x 2x 2 1 x2(1x)x ?窮?量與?

lim2 xx x

x

仍是?有限個?窮小的乘積仍是?lim⑴c=0，稱α是比β較?階的?窮小量，記為α=o(β)klimck0)c≠0k則稱α與β是k階的?

limtanxsinlim⑴c=0，稱α是比β較?階的?窮小量，記為α=o(β)k(5)limck0)c≠0k則稱α與β是k階的?

sin3

limtanxsinlim

sin3sinxsinxcos⑴c=0，稱α是比β較?階的?窮小量，記為α=o(β)；lim cos lim1cosx

sin3x x0sin2xcosxx2lim 1x0x2cos k(5)limck0)c≠0k則稱α與β是k階的?常見的等價?sin~~arcsin~tan~arctan21cos~ e1~ln(1)~a1~ln(1)1~;

1~nn1log(1)~ ln等價?因式整體替換limln(12x)sin(5x)

e231315x2arctanxcos(xcos(x 1x3

etanxesin等價?limln(12x)sin(5x)lim2x5x e2

2lim

315x2

1(5x2lim 5

arctanx2 etanxesin

x2

esinx(etanxsinx

etanxsinx x0cos(x 1x3

x0cos(x 1x3limtanxsinx

1x3

12limsinx lim(11)x

x0sin

x limf(x)limf'(x)xx0g(x) xx0g鄰域內(nèi)可導(dǎo)，且鄰域內(nèi)分母的導(dǎo)數(shù)不為0/0∞/∞

lim(x2)3x2x xlimsinx lim(11)x

x0sin

x limf(x)limf'(x)xx0g(x) xx0g0/0∞/∞

lim(x2)3x2x xlim(13)(3x2)limsinx lim(11)x

x x

x0sin

x

lim1

3(3x2) x x limf(x)limf'(x)

x3(3x2)

x xx0g(x) xx0g

ex

0/0∞/∞ 法則求極當(dāng)x→0時，下列?窮小量中，與x等價的是（ 1cos

1x22D.ex2x→0時，1ax21~則常數(shù) 法則求極當(dāng)x→0時，下列?窮小量中，sin

B. 與x等價的是（

1x2 A.1cos

1x2

C.x

2xD.2xex

ex

22

lim

1

a1x→0時，1ax21~ - 則常數(shù)

x0 1 法

x2xx22x xx

lim 1

求極 tan2x x33求極限limtan3 limx0x3 x0x3

2cos22xsin2cos2 3x2

xx25→是否可以 法則→化簡

lnlim1x→是否可以 法則→化簡

lnlim1x1ln

ln

1ln1 lim

1x

limeln

1x lnx1x

1ln1 1Qlim

lim1xlim x

ln1

x

x1

lnxlim1x1求極限limxsin(exxx1x(4)求極限limxsin(ex1limxsin(exx1

1cos(ex1)ex 21

exlim

lim x(e 1)

x1

x 11ex1lim

limex

limcos(ex1)exxx

xlim2x22x x23xlim

2tanx)cos 2x

arcsin 2x12xarctan3x21lim(1x2sinxlim2x22x

lim2(x1)(x

x23x

x1(x1)(xlimln(1tanx)cos2(2

∵x ln(1tanx) 2 cos 2 cos 22limln(1tanx)cos )2arcsin

x2

1

arctan

3x21

x2x3x2

x(2x1)(1x2) x2sin(x lim(1

lim(1x2sinx)x2sinx

x0f(x)在點x及其附近有定義f(x)在點x的極限存在f(x)在點x的極限值等于f(x)在點x的函數(shù)

若函數(shù)f(x) 設(shè)f(xx1x3

o

lima(x1)

x0

1ln(12x若函數(shù)f(x)

lim(12x)xlimex0 x02

lim

ln(12x)

lim 2

x0

x02x設(shè)f(x)x1 x3 xlim x12lim xo x x ）可去間斷點（左極限=右極限根據(jù)所計算的極限值判

f(x)

(12x)x,x0,e2x,x0.跳躍間斷 D.第?類間斷根據(jù)所計算的極限值判

f(x)

(12x)x,x0,e2x,x0.的

D.?1ln(12xlimex0limln(12x)lim2 2x0 x012xlim(e2x)e2x0

xs