当x →0时 x~sin x ~tan x ~arcsinx~arctanx~ln(1+x) ~
2. 常用极限
1. lim
n k
x →∞a x →∞
a x −1ln a
~
1+x b −1
b
(其中a >0, b≠0)
x~ex −1 x 2~1−cos x x~ −1 α~(1+x) α
2
n
11
n
=0 ,(a>1)
2. lim
c n
x →∞n! x →∞
=0, (c>0)
3. lim nq n , ( q
x →∞
n
4. lim n =1, (a>0) 6. lim
log a n n
x →∞
=0, (a>1)
1n
7. lim 9. lim
x 0 e
8. lim 1+=e
x →∞
n
x 10. lim
sin x x
x →∞
=0
) =
1p+1
11. lim
log a x x ε
x →+∞
=0 ,(a>1, ε>0)
−
n p+1
12. lim (
x →∞
1p +2p +⋯+np
n p +1
2p p+1
13. lim (
x →∞
1p +2p +⋯+np
n p +11n+1
1n+2
=212n
1
14. lim (
x →∞
1p +3p +⋯+(2n−1) p
n p +1
=15. lim
x →∞
++⋯+
=ln 2
β
17. lim (1+x) =e
x →0
1
x
16. lim
sin x x a x −1x
x →0
=1 =
αm
18. lim 20. lim 22. lim 24. lim 26. lim 28. x →0
=ln a =1 =1
19. lim 21. lim 23. lim 25. lim
(1+a)μ−1
a arcsinx x
x →0
=μ
ln(1+x)
x arctanx
x
m
x →0x →0
=1
=mn(n−m)
21
(1+mx) n −(1+nx) m
x 2
m
x →0x →0
n
− x x →0
−, (mn≠0)
n
n
∙ 1α
x m x →0
+，（mn ≠0）
n
β
x m −1x n −1
m x →1
==
m n n
, (m,n 为自然数) , (m,n ∈Z)
27. lim
x →1
m 1−x m
−
n 1−x n
=
m −n 2
x →m
29. 若Xn （n=1,2…）收敛, 则算数平均值的序列ζn= X1+X2+⋯Xn , (n=1,2⋯) 也收敛，且
n
1
lim
x1+x2+⋯+xn
n
x →∞
=lim xn
x →∞
30. 若序列Xn(n=1,2…) 收敛，且Xn>0,则lim = lim Xn
x →∞
x →∞
n
31. 若Xn>0(n=1,2…) 且lim
Xn +1
x →∞Xn
lim =lim
x →∞
n
Xn +1
x →∞Xn
32. 若整序变量Yn →+∞, 并且——至少是从某一项开始——在n 增大时Yn 亦增大，Yn+1>Yn，则
n →∞Yn
lim
Xn
=lim
Xn −Xn −1
n →∞Yn −Yn −1
4. 常用符号
5. 微分学基本公式
1. 3. 5. 7. 9.
y =c dy=0
1cos x
1. 1+2+⋯+n =
n(n+1)2
2. 12+22+⋯+n 2=
n n+1 (2n+1)
6
3. 13+23+⋯+n 3=(1+2+⋯n) 2 4. a 3±b 3= a +b (a2∓ab +b 2)
5. x n −1= x −1 (xn −1+x n −2+⋯+x +1)
6. x n −a n = x −a (xn −1+ax n −2+a 2x n −3+⋯+a 2x +a n −1) 7. x n +a n = x +a [ x 2k −1−ax 2k −2 +⋯+(a2k −2x −a 2k −1)] 8. x −1=( + +⋯+1) 9. 伯努利不等式(1+x) n ≥1+nx
n
n
1+x1 1+x2 ∙∙∙ 1+xn ≥1+x1+x2+⋯+xn
10. |x−y|≥| x −|y||
11. |xy|≥xy
12. X +X1+⋯+Xn ≥ X − X1 +⋯+ Xn 13. n!
1n+1
n+1n
) 2
1
1
14. ∙∙⋯∙
24
132n −12n
n
n
a −1n
16. 1+a
17. n −1
k
A k
18. 组合数公式C n =
k!
=
n!
k! n −k !
k k−1
C m+n+1−C mk =Cm +n+n
n!
k
排列数公式A n =n ∙ n −1 ∙⋯∙ n −k +1 =
n −k !
19. z 6−1= z +1 z −1 z 2+z +1 z 2−z +1 20. z 6+1= z 2+1 (z4−z 2+1) 21. z 4+1= z 2+ +1 (z2− +1)
1. 记号n!! 表示自然数的连乘积，这些自然数不超过n ，并且每两个数之间差2. 例：7！！=1∙3∙5∙7 8=2∙4∙6∙8
dx
2. 4. 6. 8.
y =x μ dy=μx μ−1dx y =log a x dy=
log a e x
y =a x dy=a x ln a dx y =sin x dy=cos x dx y =tan x dy=sec 2x dx =
dx
y =cos x dy=−sin x dx y =cot x dy=−csc 2x dx =
1sin x
dx
y =sec x dy=sec x tan x dx
11+x1ch 2
x
10. y =csc x dy =−csc x cot x dx 12. y =arccosx dy=14. y =arccotx dy=−18. y =cthx dy=−
11+x1sh 2x
11. y =arcsinx dy=13. y =arctanx dy=17. y =thx dy=
dx dx
15. y =shx dy=chxdx
dx
16. y =chx dy=shxdx
dx
6. 不定积分表
1. 0dx =c 3. x μdx =5.
11+x1μ+1
x a
2. 1dx =x +c 4. dx =ln |x|+c .
x 1
x μ+1+c
dx =arctanx +c
a x ln a
6.
=arcsinx +c
7. a x dx =+c 8. sin x dx =−cos x +c 10. 14. 16
1sin 2x 1sh x dx a x +x2xdx a ±x9. cos x dx =sin x +c 11.
1cos x
dx =−cot x +c
dx =tan x +c 12. sh x dx =ch x +c
dx =−cth x +c =arctan +c, (a≠0)
a
a
1
x
13. ch x dx =sh x +c 15. 2dx =th x +c
ch x 17. =ln ||+c
a −x 2a a −x
x
dx
1
a+x
1
18. 20.
=±ln |a2±x 2|+c
2
1
19. arcsin +c, (a>0)
a 21. =ln |x+ +c
± +c
x 2
22. dx = +arcsin +c, (a>0)
7. 三角学公式 sin 2θ cos 2θ
1. 基本关系
1. sin θ∙csc θ=1 3. tan θ∙cot θ=1 5. sec 2θ−tan 2θ=1 7. tan θ=
sin θcos θ
22. dx = ±
2
x
a 22
ln x + +c
2. cos θ∙sec θ=1 4. sin 2θ+cos 2θ=1 6. csc 2θ−cot 2θ=1 8. cot θ=
cos θsin θ
2. 两角和与差的三角函数公式
1. sin α±β =sin αcos β±cos αsin β 3. tan α±β =
tan α±tanβ1∓tan αtan β
2. cos α±β =cos αcos β∓sin αsin β 4. cot α±β =
cot αcot β∓1cot β±cotα
3. 倍角公式
1. sin 2α=2sin αcos α=
2tan α1+tanα
1−tan 2α1+tanα
2. cos 2α=cos 2α−sin 2α=2cos 2α−1=1−2sin 2α=
3. tan 2α=
2tan α1−tan α
=(
sin α1+cosα
2. 2
4. cot 2α=
cot 2α−12cot α
5. sin 3α=3sin α−4sin 3α 1. 3.
sin 2=
2α
1−cos α
2
6. cos 3α=4cos 3α−3cos α cos 2=
2α
1+cosα
2
4. 半角公式
=(
tan 2=
2
α1−cos α1+cosα1−cos α2
) sin α
4.
cot 2=
2
α1+cosα1−cos α
=(
1+cosα2
sin α
=(
sin α
1−cos α
) 2
5. 和差化积公式
1. 2. 3. 4. 5. 6. 7.
sin α+sin β=2sin sin α−sin β=2cos
α+β2α+β2
cos sin
α−β2α−β2
cos α+cos β=2cos
α+β2
cos
α−β2
cos α−cos β=−2sin tan α±tanβ=±cot α±cot β=±tan α±cot β=±
α+β2
sin
α−β2
sin （α±β）sin αsin β
cos （α∓β）sin αsin β
cos （α∓β）cos αsin β
6. 积化和差公式
1. 2. 3.
sin αsin β=−[cos α+β −cos(α−β)] cos αcos β=[cos α+β +cos(α−β)] sin αcos β=[sin α+β +sin(α−β)]
221121
7. 双曲函数的基本关系
1. 3. 5. 7.
cosh 2t −sinh 2t =1 coth 2t =1+sinh x =
2
1sinh 2t
2. 4. 6.
1−tanh 2t =
1cosh t
sinh 2x =2sinh x cosh x =2cosh 2x ∙tanh x cosh x =
e x +e−x
2
e x −e −x
双曲余弦的反函数
x 1+tanx
x =ln(y± y
8. 万能公式
1. 3. 5.
sin x =tan x =sec x =
2tan
2. 4. 6.
cos x =cot x =csc x =
1−tan 2x 1+tanx 1−tan 2x 2tan
2
x 1−tan x
2tan
1+tan2x 1−tan x
1+tan2x 2tan
2
当x →0时 x~sin x ~tan x ~arcsinx~arctanx~ln(1+x) ~
2. 常用极限
1. lim
n k
x →∞a x →∞
a x −1ln a
~
1+x b −1
b
(其中a >0, b≠0)
x~ex −1 x 2~1−cos x x~ −1 α~(1+x) α
2
n
11
n
=0 ,(a>1)
2. lim
c n
x →∞n! x →∞
=0, (c>0)
3. lim nq n , ( q
x →∞
n
4. lim n =1, (a>0) 6. lim
log a n n
x →∞
=0, (a>1)
1n
7. lim 9. lim
x 0 e
8. lim 1+=e
x →∞
n
x 10. lim
sin x x
x →∞
=0
) =
1p+1
11. lim
log a x x ε
x →+∞
=0 ,(a>1, ε>0)
−
n p+1
12. lim (
x →∞
1p +2p +⋯+np
n p +1
2p p+1
13. lim (
x →∞
1p +2p +⋯+np
n p +11n+1
1n+2
=212n
1
14. lim (
x →∞
1p +3p +⋯+(2n−1) p
n p +1
=15. lim
x →∞
++⋯+
=ln 2
β
17. lim (1+x) =e
x →0
1
x
16. lim
sin x x a x −1x
x →0
=1 =
αm
18. lim 20. lim 22. lim 24. lim 26. lim 28. x →0
=ln a =1 =1
19. lim 21. lim 23. lim 25. lim
(1+a)μ−1
a arcsinx x
x →0
=μ
ln(1+x)
x arctanx
x
m
x →0x →0
=1
=mn(n−m)
21
(1+mx) n −(1+nx) m
x 2
m
x →0x →0
n
− x x →0
−, (mn≠0)
n
n
∙ 1α
x m x →0
+，（mn ≠0）
n
β
x m −1x n −1
m x →1
==
m n n
, (m,n 为自然数) , (m,n ∈Z)
27. lim
x →1
m 1−x m
−
n 1−x n
=
m −n 2
x →m
29. 若Xn （n=1,2…）收敛, 则算数平均值的序列ζn= X1+X2+⋯Xn , (n=1,2⋯) 也收敛，且
n
1
lim
x1+x2+⋯+xn
n
x →∞
=lim xn
x →∞
30. 若序列Xn(n=1,2…) 收敛，且Xn>0,则lim = lim Xn
x →∞
x →∞
n
31. 若Xn>0(n=1,2…) 且lim
Xn +1
x →∞Xn
lim =lim
x →∞
n
Xn +1
x →∞Xn
32. 若整序变量Yn →+∞, 并且——至少是从某一项开始——在n 增大时Yn 亦增大，Yn+1>Yn，则
n →∞Yn
lim
Xn
=lim
Xn −Xn −1
n →∞Yn −Yn −1
4. 常用符号
5. 微分学基本公式
1. 3. 5. 7. 9.
y =c dy=0
1cos x
1. 1+2+⋯+n =
n(n+1)2
2. 12+22+⋯+n 2=
n n+1 (2n+1)
6
3. 13+23+⋯+n 3=(1+2+⋯n) 2 4. a 3±b 3= a +b (a2∓ab +b 2)
5. x n −1= x −1 (xn −1+x n −2+⋯+x +1)
6. x n −a n = x −a (xn −1+ax n −2+a 2x n −3+⋯+a 2x +a n −1) 7. x n +a n = x +a [ x 2k −1−ax 2k −2 +⋯+(a2k −2x −a 2k −1)] 8. x −1=( + +⋯+1) 9. 伯努利不等式(1+x) n ≥1+nx
n
n
1+x1 1+x2 ∙∙∙ 1+xn ≥1+x1+x2+⋯+xn
10. |x−y|≥| x −|y||
11. |xy|≥xy
12. X +X1+⋯+Xn ≥ X − X1 +⋯+ Xn 13. n!
1n+1
n+1n
) 2
1
1
14. ∙∙⋯∙
24
132n −12n
n
n
a −1n
16. 1+a
17. n −1
k
A k
18. 组合数公式C n =
k!
=
n!
k! n −k !
k k−1
C m+n+1−C mk =Cm +n+n
n!
k
排列数公式A n =n ∙ n −1 ∙⋯∙ n −k +1 =
n −k !
19. z 6−1= z +1 z −1 z 2+z +1 z 2−z +1 20. z 6+1= z 2+1 (z4−z 2+1) 21. z 4+1= z 2+ +1 (z2− +1)
1. 记号n!! 表示自然数的连乘积，这些自然数不超过n ，并且每两个数之间差2. 例：7！！=1∙3∙5∙7 8=2∙4∙6∙8
dx
2. 4. 6. 8.
y =x μ dy=μx μ−1dx y =log a x dy=
log a e x
y =a x dy=a x ln a dx y =sin x dy=cos x dx y =tan x dy=sec 2x dx =
dx
y =cos x dy=−sin x dx y =cot x dy=−csc 2x dx =
1sin x
dx
y =sec x dy=sec x tan x dx
11+x1ch 2
x
10. y =csc x dy =−csc x cot x dx 12. y =arccosx dy=14. y =arccotx dy=−18. y =cthx dy=−
11+x1sh 2x
11. y =arcsinx dy=13. y =arctanx dy=17. y =thx dy=
dx dx
15. y =shx dy=chxdx
dx
16. y =chx dy=shxdx
dx
6. 不定积分表
1. 0dx =c 3. x μdx =5.
11+x1μ+1
x a
2. 1dx =x +c 4. dx =ln |x|+c .
x 1
x μ+1+c
dx =arctanx +c
a x ln a
6.
=arcsinx +c
7. a x dx =+c 8. sin x dx =−cos x +c 10. 14. 16
1sin 2x 1sh x dx a x +x2xdx a ±x9. cos x dx =sin x +c 11.
1cos x
dx =−cot x +c
dx =tan x +c 12. sh x dx =ch x +c
dx =−cth x +c =arctan +c, (a≠0)
a
a
1
x
13. ch x dx =sh x +c 15. 2dx =th x +c
ch x 17. =ln ||+c
a −x 2a a −x
x
dx
1
a+x
1
18. 20.
=±ln |a2±x 2|+c
2
1
19. arcsin +c, (a>0)
a 21. =ln |x+ +c
± +c
x 2
22. dx = +arcsin +c, (a>0)
7. 三角学公式 sin 2θ cos 2θ
1. 基本关系
1. sin θ∙csc θ=1 3. tan θ∙cot θ=1 5. sec 2θ−tan 2θ=1 7. tan θ=
sin θcos θ
22. dx = ±
2
x
a 22
ln x + +c
2. cos θ∙sec θ=1 4. sin 2θ+cos 2θ=1 6. csc 2θ−cot 2θ=1 8. cot θ=
cos θsin θ
2. 两角和与差的三角函数公式
1. sin α±β =sin αcos β±cos αsin β 3. tan α±β =
tan α±tanβ1∓tan αtan β
2. cos α±β =cos αcos β∓sin αsin β 4. cot α±β =
cot αcot β∓1cot β±cotα
3. 倍角公式
1. sin 2α=2sin αcos α=
2tan α1+tanα
1−tan 2α1+tanα
2. cos 2α=cos 2α−sin 2α=2cos 2α−1=1−2sin 2α=
3. tan 2α=
2tan α1−tan α
=(
sin α1+cosα
2. 2
4. cot 2α=
cot 2α−12cot α
5. sin 3α=3sin α−4sin 3α 1. 3.
sin 2=
2α
1−cos α
2
6. cos 3α=4cos 3α−3cos α cos 2=
2α
1+cosα
2
4. 半角公式
=(
tan 2=
2
α1−cos α1+cosα1−cos α2
) sin α
4.
cot 2=
2
α1+cosα1−cos α
=(
1+cosα2
sin α
=(
sin α
1−cos α
) 2
5. 和差化积公式
1. 2. 3. 4. 5. 6. 7.
sin α+sin β=2sin sin α−sin β=2cos
α+β2α+β2
cos sin
α−β2α−β2
cos α+cos β=2cos
α+β2
cos
α−β2
cos α−cos β=−2sin tan α±tanβ=±cot α±cot β=±tan α±cot β=±
α+β2
sin
α−β2
sin （α±β）sin αsin β
cos （α∓β）sin αsin β
cos （α∓β）cos αsin β
6. 积化和差公式
1. 2. 3.
sin αsin β=−[cos α+β −cos(α−β)] cos αcos β=[cos α+β +cos(α−β)] sin αcos β=[sin α+β +sin(α−β)]
221121
7. 双曲函数的基本关系
1. 3. 5. 7.
cosh 2t −sinh 2t =1 coth 2t =1+sinh x =
2
1sinh 2t
2. 4. 6.
1−tanh 2t =
1cosh t
sinh 2x =2sinh x cosh x =2cosh 2x ∙tanh x cosh x =
e x +e−x
2
e x −e −x
双曲余弦的反函数
x 1+tanx
x =ln(y± y
8. 万能公式
1. 3. 5.
sin x =tan x =sec x =
2tan
2. 4. 6.
cos x =cot x =csc x =
1−tan 2x 1+tanx 1−tan 2x 2tan
2
x 1−tan x
2tan
1+tan2x 1−tan x
1+tan2x 2tan
2