极限的性质

1第四节极限的性质一．极限的性质二．极限的四则运算法则2一.极限的性质()lim()2.4,fx唯一性若存在则极性质限值唯一()lim(,()5)2.fxfx有界性若存在则在该过性质程中有界0()lim(),0,2(06).xxfxAAA保号性若且或性质则在0,()0(()0)xfxfx的某空心领域内恒有或00lim(),2.7,xxfxAx若且在的某空心领质域内性恒有()0(()0)0(0)fxfxAA或则或3二.极限的四则运算法则lim()2.4),,lim(fxAgxB如果定理则lim()2.1,,fxc若存在推为常数论则12lim(),lim(),,li2.2(m,)nfxfxfx若推都存在论lim[()()]lim()lim()fxgxfxgxABlim[()()][lim()][lim()]fxgxfxgxABlim[()]lim()cfxcfx12,,,,nccc为常数则1122lim[()()()]nncfxcfxcfx1122lim()lim()lim()nncfxcfxcfx4二.极限的四则运算法则lim()2,.3,fxAn推论若为正整数则lim2(),lim(),0.5,fxAgxBB如果且定理则lim[()][lim()]nnnfxfxA()lim()lim()lim()fxfxAgxgxB5例122lim(31)xxx求解：原式2(2)12limx2x2lim3xx2lim1x3(2)16例2.531lim232xxxx求解531lim232xxxx)53(lim)1(lim2232xxxxx.3731237解例3.321lim221xxxx求.,,1分母的极限都是零分子时x.1后再求极限因子先约去不为零的无穷小x2211lim23xxxx31lim1xxx.21)00(型消零因子法1(1)(1)lim(3)(1)xxxxx8例4.147532lim2323xxxxx求解)(型332323147532lim147532limxxxxxxxxxx.72一般,为非负整数时有和当nmba,0,000.,,,0,,lim00110110mnmnmnbabxbxbaxaxannnmmmx当当当9例5.2arctanlim22xxxxx求解.,,,0,,lim00110110mnmnmnbabxbxbaxaxannnmmmx当当当xxxxxxxxxxxarctanlim2lim2arctanlim2222221.410共扼因子法解解令61tx,变量代换法,0x,1t)1)(1()1)(1(lim1xxxxx原式)1(lim1xx.211lim321ttt原式11lim21tttt.32.11lim1xxx求例6.1111lim30xxx求例711例8).21(lim222nnnnn求解．是无穷多项之和时,n222221lim)21(limnnnnnnnn22)1(limnnnn.21先变形再求极限.12作业：习题二(A)P606.(1),(4),(6),(9),(12),(13),(16),(17),(19),(23)11.