極限的性質(zhì)及運(yùn)算法則(2)課件

1、二函數(shù)極限的性質(zhì)及運(yùn)算法則二函數(shù)極限的性質(zhì)及運(yùn)算法則M)x( fx)x(Ox0,M :xxf(x) 3 . 2def000 時(shí)有時(shí)有當(dāng)當(dāng)若若時(shí)有界時(shí)有界在在)(xfy 0 x0 x0 xxyoM-M(一一)極限的性極限的性 質(zhì)質(zhì)性質(zhì)性質(zhì)2.5.局部有界性局部有界性.Xx)x(f,A)x(flimXx某某一一鄰鄰域域內(nèi)內(nèi)有有界界過(guò)過(guò)程程所所允允許許的的在在則則稱稱若若 性質(zhì)性質(zhì)2.62.6局部保號(hào)性局部保號(hào)性B)x( f).x(g)x( fXx)x(g)x( f,BA,B)x(glim,A)x( flimXxXx 特別地特別地有有中所允許的某一鄰域內(nèi)中所允許的某一鄰域內(nèi)在在與與則則若若說(shuō)明說(shuō)明:

2、BA)B(O2 )A(O1 )B(O)x(g),A(O)x( f21 ).0)x(f(0)x(f ,x)x(Ox, 0),0A(0A,A)x(flim00 xx0 或或時(shí)時(shí)當(dāng)當(dāng)則則或或且且若若特別地特別地).0A(0A),0)x( f(0)x( f,x )x(Ox, 0,A)x( flim00 xx0 或或則則或或時(shí)時(shí)當(dāng)當(dāng)且且若若特別地特別地性質(zhì)性質(zhì)2.7 BA),x(g)x( f)(x(g)x( fXx,B)x(glim,A)x( flimXxXx 則則或或過(guò)過(guò)程程下下有有且且在在若若性質(zhì)性質(zhì)2.8(函數(shù)極限的夾逼定理函數(shù)極限的夾逼定理)A)x( flim,A)x(hlim)x(glim),

3、x(h)x( f)x(g,XxXxXxXx 則則且且所允許的某一鄰域內(nèi)所允許的某一鄰域內(nèi)若在極限過(guò)程若在極限過(guò)程 A)x(hA)x( fA)x(g x:的的某某一一鄰鄰域域內(nèi)內(nèi)有有在在說(shuō)說(shuō)明明AC利用夾逼定理可證重要極限利用夾逼定理可證重要極限(2)1sinlim0 xxx)20(, xxAOBO 圓圓心心角角設(shè)設(shè)單單位位圓圓,tan,sinACxABxBDx 弧弧于是有于是有xoBD.ACO ，得，得作單位圓的切線作單位圓的切線,xOAB的圓心角為的圓心角為扇形扇形,BDOAB的高為的高為 ,tansinxxx , 1sincos xxx即即)0 x2(也也成成立立對(duì)對(duì) ,20時(shí)時(shí)當(dāng)當(dāng) xx

4、xcos11cos0 2sin22x 2)2(2x , 02x2 , 0)cos1(lim0 xx, 1coslim0 xx得證得證. x1xlim 10 x求求例例x1x11x1 : 解解 ) , 5 . 25 . 215 . 2 , 25 . 2:( 如如1x1 xlim1x1xx1, 0 x0 x 有有當(dāng)當(dāng)1x1 xlimx1x1x1, 0 x0 x 有有當(dāng)當(dāng)1x1 xlim0 x (二二)極限的運(yùn)算法則極限的運(yùn)算法則性質(zhì)性質(zhì)2.9. 0B,BA)x(glim)x( flim)x(g)x( flim)4(;BA)x(glim)x( flim)x(g)x( f lim)3()x( flim

5、c)x(cflim )2(;BA)x(glim)x( flim)x(g)x( f lim)1(,B)x(glim,A)x( flimXxXxXxXxXxXxXxXxXxXxXxXxXx 其其中中則則設(shè)設(shè).)x( flim)x( f lim,n,)x( flim:nXxnXxXx 則則是是正正整整數(shù)數(shù)而而存存在在如如果果結(jié)結(jié)論論.5x3x1xlim)1(232x 解解)53(lim22 xxx5lim3limlim2222 xxxxx5limlim3)lim(2222 xxxxx52322 , 03 531lim232 xxxx)53(lim1limlim22232 xxxxxx.37 3123

6、 例例1 求下列極限求下列極限7x4x23x6x8lim )3(xelim )2(22xx2x )xx1(lim )5(1x1xlim )4(xn1x 消除消除0因子因子)x13x11(lim )6(21x 先通分先通分 根式有理化根式有理化 4x8xlim )7(364x tx:61 變量替換變量替換性質(zhì)性質(zhì)2.10(復(fù)合函數(shù)的極限法則復(fù)合函數(shù)的極限法則)B)y( flim)x(g( flim,B)y( flim,A)x(gy),(A)x(glimAyXxAyXx 則則或或若若.,A)x(g10. 2否則結(jié)論未必成立否則結(jié)論未必成立不能省去不能省去中條件中條件性質(zhì)性質(zhì) 意義意義1.極限符號(hào)可

7、以與函數(shù)符號(hào)互換極限符號(hào)可以與函數(shù)符號(hào)互換;.)(. 2的的理理論論依依據(jù)據(jù)變變量量代代換換xu 例例1 1.)1ln(lim0 xxx 求求. 1 xxx10)1ln(lim 原原式式)1(limln10 xxx eln 解解例例2 2.1lim0 xexx 求求. 1 )1ln(lim0yyy 原原式式解解,1yex 令令),1ln(yx 則則. 0,0yx時(shí)時(shí)當(dāng)當(dāng)yyy10)1ln(1lim 同理可得同理可得.ln1lim0axaxx 例如例如,xxysin 1sinlim0 xxx, 11sinlim nnn, 11sinlim nnn11sin1lim22 nnnnn函數(shù)極限與數(shù)列極

8、限的關(guān)系函數(shù)極限與數(shù)列極限的關(guān)系函數(shù)極限存在的充要條件是它的任何子列的極函數(shù)極限存在的充要條件是它的任何子列的極限都存在限都存在, ,且相等且相等. .sin時(shí)時(shí)的的變變化化趨趨勢(shì)勢(shì)當(dāng)當(dāng)觀觀察察函函數(shù)數(shù) xxx一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限.sin時(shí)時(shí)的的變變化化趨趨勢(shì)勢(shì)當(dāng)當(dāng)觀觀察察函函數(shù)數(shù) xxx一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限.sin時(shí)時(shí)的的變變化化趨趨勢(shì)勢(shì)當(dāng)當(dāng)觀觀察察函函數(shù)數(shù) xxx一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限.sin時(shí)時(shí)的的變變化化趨趨勢(shì)勢(shì)當(dāng)當(dāng)觀觀察察函函數(shù)數(shù) xxx一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限.sin時(shí)時(shí)的的變變化化趨趨勢(shì)勢(shì)當(dāng)當(dāng)觀觀察察函函數(shù)數(shù) xxx一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限.sin時(shí)時(shí)的的變變化化趨趨勢(shì)勢(shì)當(dāng)當(dāng)觀觀察察函函數(shù)數(shù) xxx一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限.sin時(shí)時(shí)的的變變化化趨趨勢(shì)勢(shì)當(dāng)當(dāng)觀觀察察函函數(shù)數(shù) xxx一、自變量趨向無(wú)窮大時(shí)函數(shù)的極限一、自變