一、极限运算法则定理.0,)()(lim)3(;)]()(lim[)2(;)]()(lim[)1(,)(lim,)(lim??????????BBAxgxfBAxgxfBAxgxfBxgAxf其中则设证.)(lim,)(limBxgAxf???.0,0.)(,)(???????????其中BxgAxf由无穷小运算法则,得)()]()([BAxgxf???????.0?.)1(成立?)()]()([BAxgxf???ABBA??????))((???????)(BA.0?.)2(成立?BAxgxf?)()(BABA??????)(??????BBAB.0????AB?,0,0???B?又,0???,00时当????xx,2B????????BBBB21??B21?推论1).(lim)](lim[,,)(limxfcxcfcxf?则为常数而存在如果常数因子可以提到极限记号外面..)]([lim)](lim[,,)(limnnxfxfnxf?则是正整数而存在如果推论2,21)(2BBB????,2)(12BBB???故有界,.)3(成立?二、求极限
方法
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举例例1.531lim232????xxxx求解)53(lim22???xxx?5lim3limlim2222??????xxxxx5limlim3)lim(2222??????xxxxx52322????,03??531lim232?????xxxx)53(lim1limlim22232???????xxxxxx.37?3123??小结:则有设,)(.1110nnnaxaxaxf??????nnxxnxxxxaxaxaxf?????????110)lim()lim()(lim000nnnaxaxa??????10100).(0xf?则有且设,0)(,)()()(.20??xQxQxPxf)(lim)(lim)(lim000xQxPxfxxxxxx????)()(00xQxP?).(0xf?.,0)(0则商的法则不能应用若?xQ解)32(lim21???xxx?,0?商的法则不能用)14(lim1??xx?又,03??1432lim21?????xxxx.030??由无穷小与无穷大的关系,得例2.3214lim21????xxxx求.3214lim21??????xxxx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?x)1)(3()1)(1(lim321lim1221??????????xxxxxxxxx31lim1????xxx.21?)00(型(消去零因子法)例4.147532lim2323??????xxxxx求解.,,分母的极限都是无穷大分子时??x)(型??.,,3再求极限分出无穷小去除分子分母先用x332323147532lim147532limxxxxxxxxxx?????????????.72?(无穷小因子分出法)小结:为非负整数时有和当nmba,0,000??????????????????????????,,,,0,,lim00110110mnmnmnbabxbxbaxaxannnmmmx当当当??无穷小分出法:以分母中自变量的最高次幂除分子,分母,以分出无穷小,然后再求极限.例5).21(lim222nnnnn??????求解是无穷小之和.时,??n222221lim)21(limnnnnnnnn?????????????2)1(21limnnnn????)11(21limnn????.21?先变形再求极限.例6.sinlimxxx??求解,1,为无穷小时当xx??.sin是有界函数而x.0sinlim????xxxxxysin?例7).(lim,0,10,1)(02xfxxxxxfx?????????求设yox1xy??112??xy解两个单侧极限为是函数的分段点,0?x)1(lim)(lim00xxfxx??????,1?)1(lim)(lim200??????xxfxx,1?左右极限存在且相等,.1)(lim0??xfx故三、小结1.极限的四则运算法则及其推论;2.极限求法;a.多项式与分式函数代入法求极限;b.消去零因子法求极限;c.无穷小因子分出法求极限;d.利用无穷小运算性质求极限;e.利用左右极限求分段函数极限.思考题在某个过程中,若有极限,无极限,那么是否有极限?为什么?)(xf)(xg)()(xgxf?思考题解答没有极限.假设有极限,)()(xgxf?)(xf?有极限,由极限运算法则可知:??)()()()(xfxgxfxg???必有极限,与已知矛盾,故假设错误..__________1sinlim520??xxx、.__________33lim132????xxx、一、填空题:.__________11lim231????xxx、.__________)112)(11(lim32??????xxxx、.__________5)3)(2)(1(lim43??????nnnnn、.__________coslim6??????xxxeex、练习题.__________2324lim72240?????xxxxxx、.__________)12()23()32(lim8503020??????xxxx、二、求下列各极限:)21...41211(lim1nn??????、hxhxh220)(lim2???、)1311(lim331xxx????、38231lim4xxx?????、)(lim5xxxxx??????、1412lim6?????xxx、2lim71????nmnmxxxxx、一、1、-5;2、3;3、2;4、51;5、0;6、0;7、21;8、30)23(.二、1、2;2、x2;3、-1;4、-2;5、21;6、0;7、nmnm??.练习题
答案
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