(1) lim x → 0 sin x x = 1 lim _{x ightarrow 0} frac{sin x}{x}=1 limx→0xsinx=1, 推广形式 lim f ( x ) → 0 sin f ( x ) f ( x ) = 1 lim _{f(x) ightarrow 0} frac{sin f(x)}{f(x)}=1 limf(x)→0f(x)sinf(x)=1. (2) lim x → ∞ ( 1 + 1 x ) x = e lim _{x ightarrow infty}left(1+frac{1}{x} ight)^x=mathrm{e} limx→∞(1+x1)x=e, 推广形式 lim x → 0 ( 1 + x ) 1 x = e , lim f ( x ) → ∞ [ 1 + 1 f ( x ) ] f ( x ) = e lim _{x ightarrow 0}(1+x)^{frac{1}{x}}=mathrm{e}, lim _{f(x) ightarrow infty}left[1+frac{1}{f(x)} ight]^{f(x)}=mathrm{e} limx→0(1+x)x1=e,limf(x)→∞[1+f(x)1]f(x)=e
2. 常用的等价无穷小量及极限公式(1) 当 x → 0 x ightarrow 0 x→0 时,常用的等价无穷小
(1) x ∼ sin x ∼ tan x ∼ arcsin x ∼ arctan x ∼ ln ( 1 + x ) ∼ e x − 1 x sim sin x sim an x sim arcsin x sim arctan x sim ln (1+x) sim mathrm{e}^x-1 x∼sinx∼tanx∼arcsinx∼arctanx∼ln(1+x)∼ex−1.(2) 1 − cos x ∼ 1 2 x 2 , 1 − cos b x ∼ b 2 x 2 ( b ≠ 0 ) 1-cos x sim frac{1}{2} x^2, 1-cos ^b x sim frac{b}{2} x^2(b eq 0) 1−cosx∼21x2,1−cosbx∼2bx2(b=0).(3) a x − 1 ∼ x ln a ( a > 0 a^x-1 sim x ln a(a>0 ax−1∼xlna(a>0, 且 a ≠ 1 ) a eq 1) a=1).(4) ( 1 + x ) α − 1 ∼ α x ( α ≠ 0 ) (1+x)^alpha-1 sim alpha x (alpha eq 0) (1+x)α−1∼αx(α=0).(2) 当 n → ∞ n ightarrow infty n→∞ 或 x → ∞ x ightarrow infty x→∞ 时,常用的极限公式
(1) lim n → ∞ n n = 1 , lim n → ∞ a n = 1 ( a > 0 ) lim _{n ightarrow infty} sqrt[n]{n}=1, lim _{n ightarrow infty} sqrt[n]{a}=1(a>0) limn→∞nn =1,limn→∞na =1(a>0).(2) lim x → ∞ a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 b m x m + b m − 1 x m − 1 + ⋯ + b 1 x + b 0 = { a n b m , n = m , 0 , n < m , ∞ , n > m , lim _{x ightarrow infty} frac{a_n x^n+a_{n-1} x^{n-1}+cdots+a_1 x+a_0}{b_m x^m+b_{m-1} x^{m-1}+cdots+b_1 x+b_0}=left{egin{array}{ll}frac{a_n}{b_m}, & n=m, \ 0, & nm,end{array} ight. limx→∞bmxm+bm−1xm−1+⋯+b1x+b0anxn+an−1xn−1+⋯+a1x+a0=⎩⎨⎧bman,0,∞,n=m,nm, 其中 a n , b m a_n, b_m an,bm 均不为 0 .
(3) lim n → ∞ x n = { 0 , ∣ x ∣ < 1 , ∞ , ∣ x ∣ > 1 , 1 , x = 1 , 不存在, x = − 1 ; lim n → ∞ e n x = { 0 , x < 0 , + ∞ , x > 0 , 1 , x = 0. lim _{n ightarrow infty} x^n=left{egin{array}{ll}0, & |x|1, \ 1, & x=1, \ ext { 不存在, } & x=-1 ;end{array} lim _{n ightarrow infty} mathrm{e}^{n x}= egin{cases}0, & x0, \ 1, & x=0 .end{cases} ight. limn→∞xn=⎩⎪⎪⎨⎪⎪⎧0,∞,1, 不存在, ∣x∣1,x=1,x=−1;limn→∞enx=⎩⎪⎨⎪⎧0,+∞,1,x0,x=0.(4) 若 lim g ( x ) = 0 , lim f ( x ) = ∞ lim g(x)=0, lim f(x)=infty limg(x)=0,limf(x)=∞, 且 lim g ( x ) f ( x ) = A lim g(x) f(x)=A limg(x)f(x)=A, 则有 lim [ 1 + g ( x ) ] f ( x ) = e A . lim [1+g(x)]^{f(x)}=mathrm{e}^A . lim[1+g(x)]f(x)=eA. 3. x → 0 x ightarrow 0 x→0 时常见的麦克劳林公式sin x = x − 1 3 ! x 3 + o ( x 3 ) , cos x = 1 − 1 2 ! x 2 + 1 4 ! x 4 + o ( x 4 ) , tan x = x + 1 3 x 3 + o ( x 3 ) , arcsin x = x + 1 3 ! x 3 + o ( x 3 ) , arctan x = x − 1 3 x 3 + o ( x 3 ) , ln ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 + o ( x 3 ) , e x = 1 + x + 1 2 ! x 2 + 1 3 ! x 3 + o ( x 3 ) , ( 1 + x ) a = 1 + a x + a ( a − 1 ) 2 ! x 2 + o ( x 2 ) . egin{aligned} & sin x=x-frac{1}{3 !} x^3+oleft(x^3 ight), quad cos x=1-frac{1}{2 !} x^2+frac{1}{4 !} x^4+oleft(x^4 ight),\ \ & an x=x+frac{1}{3} x^3+oleft(x^3 ight), quad arcsin x=x+frac{1}{3 !} x^3+oleft(x^3 ight), \ \ & arctan x=x-frac{1}{3} x^3+oleft(x^3 ight), quad ln (1+x)=x-frac{1}{2} x^2+frac{1}{3} x^3+oleft(x^3 ight), \ \ & mathrm{e}^x=1+x+frac{1}{2 !} x^2+frac{1}{3 !} x^3+oleft(x^3 ight),(1+x)^a=1+a x+frac{a(a-1)}{2 !} x^2+oleft(x^2 ight) . end{aligned} sinx=x−3!1x3+o(x3),cosx=1−2!1x2+4!1x4+o(x4),tanx=x+31x3+o(x3),arcsinx=x+3!1x3+o(x3),arctanx=x−31x3+o(x3),ln(1+x)=x−21x2+31x3+o(x3),ex=1+x+2!1x2+3!1x3+o(x3),(1+x)a=1+ax+2!a(a−1)x2+o(x2).
当 x → 0 x ightarrow 0 x→0 时,由以上公式可以得到以下几组“差函数”的等价无穷小代换式:
x − sin x ∼ x 3 6 , tan x − x ∼ x 3 3 , x − ln ( 1 + x ) ∼ x 2 2 x-sin x sim frac{x^3}{6}, quad an x-x sim frac{x^3}{3}, quad x-ln (1+x) sim frac{x^2}{2} x−sinx∼6x3,tanx−x∼3x3,x−ln(1+x)∼2x2, arcsin x − x ∼ x 3 6 , x − arctan x ∼ x 3 3 arcsin x-x sim frac{x^3}{6}, quad x-arctan x sim frac{x^3}{3} arcsinx−x∼6x3,x−arctanx∼3x3.
4. 基本导数公式( x μ ) ′ = μ x μ − 1 ( μ 为 常 数 ) , ( a x ) ′ = a x ln a ( a > 0 , a ≠ 1 ) , ( log a x ) ′ = 1 x ln a ( a > 0 , a ≠ 1 ) , ( ln x ) ′ = 1 x , ( sin x ) ′ = cos x , ( cos x ) ′ = − sin x , ( arcsin x ) ′ = 1 1 − x 2 , ( arccos x ) ′ = − 1 1 − x 2 , ( tan x ) ′ = sec 2 x , ( cot x ) ′ = − csc 2 x , ( arctan x ) ′ = 1 1 + x 2 , ( arccot x ) ′ = − 1 1 + x 2 , ( sec x ) ′ = sec x tan x , ( csc x ) ′ = − csc x cot x , [ ln ( x + x 2 + 1 ) ] ′ = 1 x 2 + 1 , , [ ln ( x + x 2 − 1 ) ] ′ = 1 x 2 − 1 egin{array}{ll} left(x^mu ight)^{prime}=mu x^{mu-1} ( mu 为常数), & left(a^x ight)^{prime}=a^x ln a(a>0, a eq 1), \ \ left(log _a x ight)^{prime}=frac{1}{x ln a}(a>0, a eq 1) , & (ln x)^{prime}=frac{1}{x}, \ \ (sin x)^{prime}=cos x, & (cos x)^{prime}=-sin x, \ \ (arcsin x)^{prime}=frac{1}{sqrt{1-x^2}}, & (arccos x)^{prime}=-frac{1}{sqrt{1-x^2}}, \ \ ( an x)^{prime}=sec ^2 x, & (cot x)^{prime}=-csc ^2 x, \ \ (arctan x)^{prime}=frac{1}{1+x^2}, & (operatorname{arccot} x)^{prime}=-frac{1}{1+x^2}, \ \ (sec x)^{prime}=sec x an x, & (csc x)^{prime}=-csc x cot x, \ \ {left[ln left(x+sqrt{x^2+1} ight) ight]^{prime}=frac{1}{sqrt{x^2+1}},}, & {left[ln left(x+sqrt{x^2-1} ight) ight]^{prime}=frac{1}{sqrt{x^2-1}}} end{array} (xμ)′=μxμ−1(μ为常数),(logax)′=xlna1(a>0,a=1),(sinx)′=cosx,(arcsinx)′=1−x2 1,(tanx)′=sec2x,(arctanx)′=1+x21,(secx)′=secxtanx,[ln(x+x2+1 )]′=x2+1 1,,(ax)′=axlna(a>0,a=1),(lnx)′=x1,(cosx)′=−sinx,(arccosx)′=−1−x2 1,(cotx)′=−csc2x,(arccotx)′=−1+x21,(cscx)′=−cscxcotx,[ln(x+x2−1 )]′=x2−1 1 三角函数六边形记忆法:
注: 变限积分求导公式. 设 F ( x ) = ∫ φ 2 ( x ) φ 1 ( x ) f ( t ) d t F(x)=int_{varphi_2(x)}^{varphi_1(x)} f(t) mathrm{d} t F(x)=∫φ2(x)φ1(x)f(t)dt, 其中 f ( x ) f(x) f(x) 在 [ a , b ] [a, b] [a,b] 上连续, 可导函数 φ 1 ( x ) varphi_1(x) φ1(x) 和 φ 2 ( x ) varphi_2(x) φ2(x) 的值域在 [ a , b ] [a, b] [a,b] 上, 则在函数 φ 1 ( x ) varphi_1(x) φ1(x) 和 φ 2 ( x ) varphi_2(x) φ2(x) 的公共定义域上有: F ′ ( x ) = d d x [ ∫ φ 1 ( x ) φ 2 ( x ) f ( t ) d t ] = f [ φ 2 ( x ) ] φ 2 ′ ( x ) − f [ φ 1 ( x ) ] φ 1 ′ ( x ) . F^{prime}(x)=frac{mathrm{d}}{mathrm{d} x}left[int_{varphi_1(x)}^{varphi_2(x)} f(t) mathrm{d} t ight]=fleft[varphi_2(x) ight] varphi_2^{prime}(x)-fleft[varphi_1(x) ight] varphi_1^{prime}(x) . F′(x)=dxd[∫φ1(x)φ2(x)f(t)dt]=f[φ2(x)]φ2′(x)−f[φ1(x)]φ1′(x).
5. 几个重要函数的麦克劳林展开式(1) e x = 1 + x + 1 2 ! x 2 + ⋯ + 1 n ! x n + o ( x n ) mathrm{e}^x=1+x+frac{1}{2 !} x^2+cdots+frac{1}{n !} x^n+oleft(x^n ight) ex=1+x+2!1x2+⋯+n!1xn+o(xn).
(2) sin x = x − 1 3 ! x 3 + ⋯ + ( − 1 ) n 1 ( 2 n + 1 ) ! x 2 n + 1 + o ( x 2 n + 1 ) sin x=x-frac{1}{3 !} x^3+cdots+(-1)^n frac{1}{(2 n+1) !} x^{2 n+1}+oleft(x^{2 n+1} ight) sinx=x−3!1x3+⋯+(−1)n(2n+1)!1x2n+1+o(x2n+1).
(3) cos x = 1 − 1 2 ! x 2 + 1 4 ! x 4 − ⋯ + ( − 1 ) n 1 ( 2 n ) ! x 2 n + o ( x 2 n ) cos x=1-frac{1}{2 !} x^2+frac{1}{4 !} x^4-cdots+(-1)^n frac{1}{(2 n) !} x^{2 n}+oleft(x^{2 n} ight) cosx=1−2!1x2+4!1x4−⋯+(−1)n(2n)!1x2n+o(x2n).
(4) 1 1 − x = 1 + x + x 2 + ⋯ + x n + o ( x n ) , ∣ x ∣ < 1 frac{1}{1-x}=1+x+x^2+cdots+x^n+oleft(x^n ight),|x|