大一上高數(shù)b1老師版

2/ ln(1+ x→0 ?1)tan√

1+x?e3ln(1+ etanx?esin x→0(x+x2)ln(1+x)arcsin 3/若limf(x)=0,x→x

=o(1)(x→x0)..4/設(shè)x→x0時(shí),u(x)v(x)為無(wú)窮小limu(x)0,u(x)是v(x)的高階無(wú)窮小,vx→x0vu(x)

o(

limx→0

=0,x2=o(x)limx→0

=

x3=o(x)5/limu(x)=1,······等價(jià)無(wú)窮小,u(x)～v(x)(xx),x→x0

u(x)=v(x)+o(v(x))(x→x0).例如

=1,sinx～x(x→0),sinx=x+o(x)(x→0).6/limu(x)=C?=0,······同階無(wú)窮小,x→x0u(x)=O(v(x))(x→x0), 或u(x)～Cv(x)(x→x0).例如

1?cosx

1,1?cosx

1

(x0),

o =1 2 ()(→0).當(dāng)x→x0,u(x)→0時(shí),常取v(x)=(x? 7/定義若limf(x)=∞,x→x

時(shí),f(x)是無(wú)窮大量設(shè)x→x0時(shí),u(x)v(x)為無(wú)窮大量limu(x)=∞,u(x)是v(x)的高階無(wú)窮大x→x0limu(x)=1,······等價(jià)無(wú)窮大,記為u(x)～v(x)(x→x limu(x)=C?=0,u(x)是v(x)的同階無(wú)窮大x→x0,設(shè)a>1,p>0 ,

=+∞,

=+∞x→+∞ x→+∞ln 8/例.設(shè)f(xx2x5.x→∞,f(x)～1例.設(shè)g(xx√x.1

x→0,f(x)～1x→+∞,g(x)～x21結(jié)論:設(shè)u(xu1(x

x→0,g(x)～x4u(x為無(wú)窮大量ui(x等價(jià)u(x為無(wú)窮小量,uj(x等價(jià)9/等價(jià)量:等價(jià)無(wú)窮小量或等價(jià)無(wú)窮大量.當(dāng)x→0時(shí),常用的等價(jià)無(wú)窮小:sinx～x,ln(1+x)～x,ex?1～x,(1+x)α?1～1?cosx

1x2,tanx～x,arcsinx～x,arctanx～210/定理6.4(積與商的代換u(x)v(x)w(x)在x

的某個(gè)去心鄰域上有定義,且limv(x)=x→x0即v(x)～w(x)(x→x0),limu(x)v(x)=lim limv(x)=limw(x)x→x0 limu(x)=limu(x)x→x0 x→x0等價(jià)代換的目的:為了化簡(jiǎn)將積或商中表達(dá)式復(fù)雜的因子換11/例如:(1)

ln(1+x→0(e2x?1)sinx(2)

31+tan2x?

sin(x12/問(wèn)6.5和與差的等價(jià)代換問(wèn)題u(x)?求

.假設(shè)u(x)～u1(x),v(x)～

1(x)(x→x

limu(x)? =

u1(x)?

(?) √

例如:(3)

1+x?eln(1+ (4)limtanx?sinx 13/limlimu(x)???=limu1(x)?在以下過(guò)程中,情形1. limu?v

=lim

ww–limww等.. =limw?lim=limu1 w等式成立的條件:limu1,limv1都存在例如:

√ 31+x 3le(1 14/limlimu(x)?=limu1(x)?

u?v～u1?v,1即 u?v

=u1?limu?v?(u1?v1)=u1?=u?u1?v?v1=u/u1?1?v/v1?1u1? u1? 1? u1/v1?分子中,u/u1?1,v/v1?1→0.欲使整個(gè)式子極限為零,極限非零,即limu1=:設(shè)u～u,v～v.當(dāng)muv

1時(shí),u?v～u1?v15/定理6.6和與差中的等價(jià)代換求limu(x?v(x)時(shí),設(shè)下面兩個(gè)條件之一成立 (1)limu(x) limv(x)都存在 (2)

u(x)?=1.x→x0 x→x0 x→x0u～u1,v～v1,limu(x)?v(x)=limu1(x)?v1(x).√

例如:

1+x le(1

,(1)(2)都滿足limtanx?sinx,(1)(2都不滿足 16/tanx

sinx(1?cos

x·1 lim =

=lim =

x3cos

.tanx～x,sinx～將得到錯(cuò)誤的結(jié)果.tanx=x+o(x),sinx=x+o(x),limtanx?sinx=limo(x)=? tanx～x,sinx～x,tanx?sinxx的高階無(wú)窮小.但具體是2階,還是3階,我們并不清楚.17/作業(yè)18/定理6.7(無(wú)窮小量的運(yùn)算當(dāng)x→0時(shí),用o(x)表示比x高階的無(wú)窮小量 o(xm)+o(xn)=o(xm)(mo(xn)±o(xn)=xm·o(xn)=o(xm)·o(xn)=k為常數(shù)時(shí),k·o(xo(xφ(x有界時(shí),φ(x·o(xn 19/

etanx?esin.2.x→0(x+x)ln(1+x)arcsin分?.當(dāng)x→0時(shí),x+x2=x(1+x)～x,ln(1+x)～x,arcs