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1
1−1
1.xlimf(x,y)D→x
y→y0
0
x∈D
inff(x)+infg(x)inf{f(x)+g(x)}inff(x)+supg(x)
x∈D
x∈D
x∈D
x∈D
2.f(x)=x−[x](−∞,+∞)13.
4.S⊂RS5.f(x)If(x)I6.f(x)(−∞,+∞)k>0,T>0
f(x+T)=kf(x)aTϕ(x)f(x)=axϕ(x).
7.f(x),x∈(−∞,+∞)x=ax=b
f(x)
8.f:R→RRA,B⊂R
(1)A⊂f−1[f(A)];
(2)f−1[A∪B]=f−1(A)∪f−1(B)(3)f−1[A∩B]=f−1(A)∩f−1(B)C
(4)f−1AC=f−1(A).
9.AαB={x+α|x∈A}
supB=supA+α,infB=infA+α.
10.f(x)=sinx211.f(x)=x2+ax+b
1
max{|f(0)|,|f(1)|,|f(−1)|}.
212.f(x)R2f(x)+f(1−x)=x2,x∈R,
f(x)
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21n+1.a>1,an=+···+n,n=1,2,···,liman.
n→∞aa2a
111
2.lim1−21−2···1−2.
n→∞23n
112
3.lim1−1−···1−.
n→∞36(n+1)(n+2)
n3−123−133−1
···3.4.lim3n→∞2+133+1n+11
5.|an+2−an+1|<|an+1−an|,n1.:{an}
2
111
6.lim1+1+++···+=e.
n→∞2!3!n!111
7.an=1+++···+−lnn,n=1,2,c(cEuler).
23n
111
8.lim++···+.
n→∞n+1n+22naann
inf.9.0am+nam+an,m,n=1,2,···,lim
n→∞nn∈Nnan(a2n+3a),a>0.liman10.a1>0,an+1=
n→∞3a2+an
1
11.a1=1,an+1=1+,n=1,2,···,{an}
an1
12.a0=1,an=,n=1,2,···,liman.
n→∞1+an−1
13.lim(a1+a2+···+an)
n→∞
1−2
(1)lim
1
(a1+2a2+···+nan)=0;n→∞n1(2)lim(n!·a1·a2···an)n=0.
n→∞
12kn
a0+Cna1+Cna2+···+Cnak+···+Cnan
14.liman=a,lim=a.
n→∞n→∞2n1p+3p+···+(2n−1)p
.15.lim
n→∞np+116.0 n→∞ n→∞ n→∞ 2n a.1−λ http://shuxuefenxi.ys168.com/3 1−3 √ 1.2.3.4.5.6.7. √√ x−a+x−a√lim.22x→ax−a√√m 1+αx·n1+βx−1lim.x→0x√√√(1−x)(1−3x)···(1−nx)lim.n−1x→1(1−x) n (x+a1)(x+a2)···(x+an)−x.lim x→+∞ √√nn 22x−x−1+x+x−1lim.x→+∞xn√n√n 221+x+x−1+x−x lim.x→0x(1+x)α−1 (α=0).lim x→0ln(1+x)x→+∞ 8.lim[(x+a)α−xα],α>0,α=1,a>0. f(x) α=1,f(x)−f(αx)=o(x)(x→0), x→0xf(x)=o(x)(x→0). x 11 10.limsin+cos. x→∞xx9.lim xα−aα (a>0).11.limβx→ax−aβx+11x+1x+1xa+b+c 12.lim(a>0,b>0,c>0). x→0a+b+c12xxx2a+b (a>0,b>0).13.lim x→0ax+bxaa−ax 14.limx(a>0). x→aa−xa15.f(x)(0,+∞)f(x)0,lim ∀α>0,lim f(αx) =1. x→+∞f(x)f(2x) =1, x→+∞f(x)x a http://shuxuefenxi.ys168.com/4 1−4 atx 1.f(x)[a,b]M(x)=supf(t),x∈[a,b], M(x)[a,b]. 2.x,y∈Rf(x+y)=f(x)+f(y) f(x)=ax. 3.f(x)R (1)f(0+0),f(0−0) (2)x,y∈R, x+yf(x)+f(y)f= 22f(x)=[f(1)−f(0)]x+f(0),x∈R.4.f(x)x=0 f(αx)=βf(x),α>1,β>1. f(x)x=0. 5.f(x)R (α,β), S={x|f(x)∈(α,β),x∈R} . 6.f(x)[a,b][a,b] f(x)[a,b]. 7.f(x)RA,B, ∀x∈R,|f(x)|A|x|+B. 8.f(x)[a,+∞)limf(x) f(x)[a,+∞). x→+∞ http://shuxuefenxi.ys168.com/5 9.f(x)I ∀{xn},{xn}⊂I,lim(xn−xn)=0⇒lim[f(xn)−f(xn)]=0. 10.f(x)[a,+∞)g(x)[a,+∞) lim[f(x)−g(x)]=0.g(x)[a,+∞). 11.f(x)g(x)[a,b]f(x){xn}⊂[a,b], ∀n∈N,g(xn)=f(xn+1).∃x0∈[a,b],f(x0)=g(x0). 12.f(x)[0,+∞)∀c∈R,f(x)−c limf(x). 13.f(x)(a,+∞) ∀T,∃{xn}⊂(a,+∞),limxn=+∞, n→∞ n→∞ x→∞ x→+∞ n→∞ n→∞ lim[f(xn+T)−f(xn)]=0. x→∞ 14.f(x)g(x)Rlim[f(x)−g(x)]=0. f(x)=g(x),x∈R. 15.y=x+[x]. 16.y∈R,cotx=yx(0,π)x=x(y), x=x(y). 17.ωf(δ)= f(x)(a,b)limωf(δ)=0.+ δ→0 |x1−x2|δ sup|f(x1)−f(x2)|,x1,x2∈(a,b), http://shuxuefenxi.ys168.com/6 1. ToeplitzStolz t11t21t31···tn1··· t22t32tn2 t33 ··· tnn·········tn3··············· n tnk=1,n=1,2,3,···,1;(1) (2)k,limtnk=0,.Toeplitz.{tnk}2. {tnk}α1,α2,α3,...,αn,... βn=tn1α1+tn2α2+···+tnnαn,n=1,2,... α1,α2,α3,...,αn,...β1,β2,β3,...,βn,.... 1{tnk},a1,a2,a3,...,an,... bn=tn1a1+tn2a2+···+tnnan,n=1,2,... b1,b2,b3,...,bn,...a1,a2,a3,...,an,.... liman=0limbn=0. n→∞ n→∞ n→∞ k=1 http://shuxuefenxi.ys168.com/7 ε∀ε>0,∃m∈N,k>m,|ak|<2. mp∈Nn>ptn1|a1|+tn2|a2|+···+tnm|am|< ε2((2)). N=max{m,p}n>N |βn|tn1|α1|+···+tnm|αm|+tnm+1|αm+1|+···+tnn|αn|< ε2ε+(tnm+1+···+tnn)2 ε2+ ε2=ε. {tnk},{un}a, nvn=tnkuk,n=1,2,···limvn=a k=1 n→∞ limun=aun=a+αn,{αn} n→∞ vn==a nk=1 nk=1 tnk(a+αk)tnk+ nk=1 tnkαk =a+ n→∞ nk=1 nk=1 tnkαk. tnkαk limvn=a Stolz{xn}{yn}0 n→∞ ynxn→∞n =a. yn−yn−1 n→∞xn−xn−1 =a, x0=y0=0 tnk= xk−xk−1 ,n=1,2,···,k=1,2,···,n.xn yn−yn−1,xn−xn−1 un= vn== n n=1,2,··· nxk−xk−1k=1 tnkuk= 1 xn k=1 nk=1 xnyn xn · yk−yk−1xk−xk−1 (yk−yk−1)= n→∞ limvn=alim ynxn→∞n =a. 因篇幅问题不能全部显示，请点此查看更多更全内容