大一上高数b1老师版

;:;;);;3/

Cauchy收敛原理

,4/习题lim

=n→∞n习题2(√

1+k−15/

)111

= 1x→∞

=e⇒lim(1+x)x=幂指函数的极限:设limf(x)=A,limg(x)=B.求limA>0,且AB均为有限常数,limf(x)g(x)=limeg(x)lnf(x)=elimg(x)lnf(x)=eBlnA=若A=1,B=∞且lim(f(x)−1)g(x)=C,f(x)limf(x)g(x)=lim[(1+f(x)−1)1](f(x)−1)g(xf(x)6/limxn=n→∞

limx2k−1=limx2k=k→∞ k→∞limx=A

limx1+x2+···+xn=n→∞ n→∞ √ n→∞

nx1x2···xn=A(xn>√limxn+1=A√

limnxn=A(xn>习题

n→∞

n→∞设{x2k−1},{x2k},{x3k}都收敛.试问:{xn}是否收敛7/其它结论.设p0,a

p

= n= = n=n→∞ n→∞ n→∞ n→∞

p

= x=x→+∞ |x|< |x|>

x→+∞x→0+,1→+xlimxnn→∞

x=

x→0—

1→−x x=

求类似lime11

8/常用的等价无穷小:当x→0时sinx,

(1

ex

1∼

(1+x)α

1∼α

1−cosx

2等价代换的两种形式:u∼v,即u=v+9/【1】0,∞. 0∞的因子【2】∞·0,∞−∞.∞·0,对lim=0的因式用等价代换∞−

⇒∞⇒∞

0,∞ uv=evln

0,∞.3】1,0, =======⇒0·∞,∞−∞对1∞型,凑重要极限或者用幂指数运算法则. 10/习题

sin2

习题() ()lim√ √

lim

3x− 1+xsinx cos

x→∞习题5 习( 1−cos

x2−2xlim √

— x→0x(1

x+ x+

1+cos

)2x—

习题 lim .

lim

ln(1+

x→∞

x+11/

)

习题 + lim(cotx)lnxx→∞ 习题lim(arcsinx)tanx.

131+1−

.12/ 例.

2,

=2+x.求limx

n→∞13/后在递推式两边取极限,最后解方程A=f(A).P26,例2.10.

2,

=2+x.求limx

n→∞P31,B3.x0=2,xn+1=2+1.求lim n→∞14/x=f(x)有解x=A,xn=A+αn,A+αn=f(A+再由A=f(A)得αn=f(A+αn−1)−f(A).然后用适当放缩法证明αn→0.P26,例2.10.

2,

=2+x.求limx

n→∞P31,B3.x0=2,xn+1=2+1.求lim n→∞15/ A=f(A)是恒等式,则前面的方法失效.此时根据递推关系写出数列的通项表达式,再求极限.P31,习题设

=a,

=b,

=xn+xn−1,求limxn n→∞n,习题已知abx1x2>0,xn+1=axn+

(n2).求limxnn→∞16/17/习题limxm−1,m,

习题lim(xn−an)−nan−1(x

xn ∈

(x−a) 习题lim(1+x)(1+2x)(1+3x)−1

—n 1 118/ 习题

1−x−2+3

limx→0

1+5x−(1+x)习题√21 √

√ √

327+x−327−.x+23

limm1+αxn1+βx−. .√22 √ x+2

3x+.

x+9−

1 1 19/习题lim(1

x)···(1

√n

(1− 习题27√ √ x→+

(x x2−1)n+(x x2−1) lim

1+x2+x)n−x

1+x2−.20/√ √ lim√1+x−√1−x

习题32 x2 x

31+x

31−

x→+

4x3+x−习√

√

习√33 31+x

41+

x−√a

x−

3√ 21 1−2

x2−

习题(34 m1+mx−n1+

x x+√x−√

x→+

21/习题

√

√

u1 − 1 1 习题[√

]x→+

n(x+a1)(x+a2)···(x+an)−x(习题(√

√ (a+x)(b+x) (a−x)(b−x)x→∞22/习题limsin

习题 ( 4limtan2x −x4x→πsin习题 sinx−

3x→3

.1−2cos.

.√ cosx2

3cos.

习题lim(1+x2)cot2 sin习题

习题lim(1−x)tanπx

lim(sinx)tan

23/

)

习题ax− + x→∞

x−习题

习题ax2−limx[ln(1+x)−ln

x

x2,a,b>x→+习题

x→0(a−b习题

ln(2+

lim

π−arctanx→+∞ln(3+

x→∞

x+ 习题

[5]lntanπ+ lim .

24/习题c0,任取0x0<1,cxn+1=xn(2− (n=0,1,2,···求limn→∞25/习题x=qsinx+a,(0q1,a是常数.x1=qsinx0+a,···,xn=qsinxn−1+a,··证明:{xn}的极限存在且恰好 26/习题设fn(x)=cosx+cos2x+···+cosnx.证明∀n∈Z,f(x)=1在[0,π)中有且仅有一个正根 3