1、當XTO時x-ex - 1"lcosx?x»l + x 1 a-(l + x)a1 常用等價無窮小a - 1 (1 + x) -1xsin xtan xarcsinxarctanxln (1 +其中a > 00 豐 0)2.常用極限I.lim-= 0 Aa > 1)x-*o>a2.liin= 0,(c > 0)X->8 *5.7.9.limnqr(|q| < 1)XTS1X-*slim 話=4.10.liinS = lr(a > 0)X>oolhn=O,(a>l)X-KW 11lim學=0K->8）二而lim -=

2、Or(a> 1,F> 0)lim (Xt + SX12.x-81P+ 2P +npn1Inn (p+ip + /2X-*oon14.lim (n+rT2 + - + ) = ln211.13.吠7;:嚴巧諾x-*o>n尸28.16.lim=1lO xlim(l + x)x = e17. x-0ax-1,(l + a/- 1lim= Inalim=卩18.x-*0 %19.x-»0,ln(l 4x)arcsinxlim-= 1lim = 120.zO x21.x-0 x22.arctanxlim= 1zO x23.(1 + nix)- - (1 + nx) 1 lim

3、:= rmix-*0xz1 + ax+ Bx a Plim二一一-(inn * 0)/L + ax 71 t- px la 卩 hmxm +24.lOxm n25.lOxmhm “ 二，（m,n為 IJ 然數）.f mn mnh叫、m , a = 226.x 十-1n27.X-l1 - X /Z15.(mn*O)= -,(m,neZ)29.若如（n=12）收斂，則算敵平均值的序列S=dXl+X2 + .X】）,（n=：l,2）也收斂，且xl 4-X2 + .+ xnlim= linixnnlim 寸xl + x2 + + xn = limXn若序列Xa(n=12)收斂，旦XnX),則x心x-&

4、gt;<»u若沁Z/吋社在，則娜丘豐專32.若整序變雖Yiit + 8,芥且至少是從某一項開貼在門喈大時丫門亦増大，Yn+1>Yn,則Xn hm石 n83常用公式及不等式1n(n+ 1)2.W + . + n2二怦竺巴10.12.13.zn +1 n n! < ()1 3 2n-l 1：14.2 42n 、伽 + 115.16.1 +a<e17.18.徘列敎公式A："】©（nk+l） =召19.;6-l =(z+ l)(z- 1)(/ + z+ l)(z2-z) + l20.z6 + l = (z2+l)(z4-z2 + l)21.z1 +

5、 1 = (z2 + 善z + l)(z2+ 1)4.常用符號1記號述表不自然數的連乘積，這些自然數不超過4并且苒悶個敵之間差2.例：7! ! = 1-3-5-78! = 24683. 13 + 23 + . + n3 = (1 + 2 + .n)24. a3 ± b3 = (a + b)(a2 + ab + b2)5. xn- 1 = (x- l)(xn_1 + xn_2+ . + x+ 1)6. xn - a11 = (x - a)(xn 1 + axn-2 + a2xn-3 +. + a2x + an'l)7 xn + an = (x + a)(x2k_1-ax2k_2

6、)+ . + (a2k_2x-a2k_1)|& x-1=(農E +陰J.+ 1)9.伯勞利不等式(1 + x)n > 1 + nx(1 + xl)(l + x2)(1 + xn) > 1 + xl + x2 + . + xn|x-y|>|x|-|y|ii. |xy|>xy|X + XI + + Xn| > |X|- (|X1| + . + |Xn|)5微分學基本公式1.y = cdy = 03y = asdy = axln adx45.y = sin xdy = cos xdx7.y = tan xdy = sec2 xdx = v-dx曲x8.9.y =

7、 sec xdy = sec xtan xdx11.y = arcsinx斫掙13.y = arctanxdy = #15.y = shxdy = clixdx17.y = tlixdy=4-dxch xy =legdy牛6.y = cos xdy =- sin xdxy =cotxdy =- esc2 xdx = j-dx sin x10.y = esc xdy =- esc xcol xdx12.y = arccosx dy =- -ydx14.y = arccotxdy =- -?dx16.y = chxdy =shxdx18.y = cthxdy=-4-<ixsh x2. y =

8、 xdy = pxM" dx6不定積分表Jodx = c1.5.= arctaiix + cJldx二 x + c2.21.冷士Rc用dx二訂+c&/sin xdx =-cosx + c/cos xdx = sin x + c10.fr-dx =- cot x + c sin x= tan x + cCOS X12.fsh xdx = ch x + cJchxdx 二 shx + c14.fdx= - cth x + c sh-x/y-dx= th x + c ch x16r dx1x/c、J-= -arctan- + c,(a 豐 0) a +xaA去二瓠啓+c18.J呂之

9、刼|a2±0 + ca ±x7.9.11.13.15.17.=arcsine + c,(a > 0)a19.20.In |x + Jx* ±a2| +c22 f Ja2 - x2dx =-x2 + arcsin + c,(a > 0)22.± a?dx =眼 士 / +1-1 n |x + 'x2±a2| + c7.三甬學公式sin20 cos20L基本關系sin B esc 6 = 11.2.cos 0 sec 9 = 15.7.tan 0 cot 0 = 1in2 0 + cos2 0 = 1sec2 0 - tan2

10、0 = 16.csc20-cot20 = lA CgOcot0 = e2.阿角和與羞的三角函敵公式I sin (a ± P) = sin acos 0 士 cos asin P2.cos (a ± 卩)二 cos acos 0 + sin asin pz , c、 tdna 土忖np13,1 ± 0)= iTtanatanPcot acot p=F 1 cot(a±p)= fotp±fota3 倍角公式1.,r r ,2Un<xsin 2a = 2sm acos a =l -t- tan a2.cos 2a = cos2 a - sin2

11、 a = 2cos2 a - 1 = 1 - 2sinZl - tan4 aa=r72tan atan 2a =1 - tan acot2 a- 1COt 2(X = F廠5.sin 3a = 3sina-4sin a6.cos 3a = 4cos a - 3cos a4半角公式.2«1 - cos a1.sin22 a1 - cos «zl-sin a3tan -=1 + cos a-< sina) Q 十 cos2 a1 + cos aA +sina4.cot 2 =1 - cos a=() K sin a 丿"- cos2.a)2 a)2 a 14 c

12、os a COS x = z5和差化積公式1.sin a + sin 0 = 2sin -|A:os 乎c小a+卩.a-卩sin a - sin 0 = 2cos -sin cos a + cos p = 2cos Aos 乎a +卩a -卩cos a - cos 3 = 2sin -sin 5.sin (a ±p)tana±tan3=±p-2.4.cota±cotp=±cos (aT p)sin asin Rco$ (a T p>7. tan at cot3 =± fosasinp 6.枳化和差公式sin asin 0 =-字cos (a + 0) - cos (a - B)cos acos B = |cos (a + 0) + cos (a - 0)sin acos 0 = ；sin (a + B) + sin (a - p)7 雙曲函敵的基本關系1 cosh21-sinh21 = 1_ e -e sinnx = 1 - tanh21 =y-2.coshz t24. sinh 2x = 2sinh xcosh x = 2cosh x tanh xX . -K - e +e , cosh x = 7.雙曲余弦的反函敎x = In (y