数学分析函数的连续性-2-5

§2.5函数的极限(2)2008／10／23四、极限存在定理夹逼定理.1,)(0时如果当xUxo),()()()1(xhxfxg,)(lim,)(lim)2(00AxhAxgxxxx.,)(lim0Axfxx且等于存在那么)||(Mx或)(x)(x)(x证明：;)(,||0,0101AxgAxx时当时当取||0),,min(021xx;)()()(AxhxfxgA;)(,||0,0202AxhAxx时当.|)(|Axf故,0.)()(.),(),,,(000时的子列当为函数称数列则时使得中有数列或可以是设在过程axxfxfaxnaxxxxaaxnnn定义.)(lim)},({)(,)(limAxfxfxfaxAxfnnnax都有子列的任何时Heine定理(函数极限与数列极限的关系)定理Heine.2证.)(,0,0,00Axfxx恒有时使当Axfxx)(lim0.0,,0,00xxNnNn恒有时使当对上述,)(Axfn从而有.)(limAxfnx故,lim00xxxxnnn且又不成立，假设Axfxx)(lim0,00则必,N*n对的存在满足nxx1||00.0|)(|0Axfn使得，点nx},|{0xxxnn即找到了一个数列,0xxn但是.)(limAxfnn函数极限存在的充要条件是它的任何子列的极限都存在,且相等.例如,xxysin1sinlim0xxx,11sinlimnnn,11sinlimnnn11sin1lim22nnnnnxy1sin例7.1sinlim0不存在证明xx证,1nxn取,0limnnx;0nx且,2141nxn取,0limnnx;0nx且nxnnnsinlim1sinlim而,1214sinlim1sinlimnxnnn而1limn二者不相等,.1sinlim0不存在故xx,0存在)(lim0xfxx,||0,||00201xxxx.)()(21xfxf都有证明：,)(lim0Axfxx.2|)(|,00Axfxx时当则特别：取2,1,||00ixxi.22)()()()(2121AxfAxfxfxf,0,0)(.3Cauchy柯西准则如满足使得,,0,0,21xx①任取一个)(lim},{00xxxxxnnnn,N0,*N对,0,0xxNnmn时，当||00xxm|)()(|mnxfxf.)}({列是Cauchyxfn.)(lim存在xnnlxf则),(lim},{00xyxyynnnn.)}({列也是Cauchyyfn.)(lim存在ynnlyf另一个②③往证：yxll),(lim00xzxznnn且.)(limlzfnn}.{,,,,,,,2211nnnzyxyxyx组成将,},{00xxxxnn的数列对任意收敛于.)(limlxfnn都有知定理据,Heine.)(lim0lxfxx,}{}{},{的子列是由于nnnzyx.lllyx五、函数极限的性质.)(lim0存在，则极限必唯一若xfxx证明：定理，由任取Heinexxxxnn)(00.),(lim)(lim0由数列极限唯一性可知xfxfxxnn1.唯一性证明：1.|)(|||0,01,0Axfxx时取.||1|||)(||)(||)(|MAAAxfAAxfxf.)(,)(lim000内有界的邻域则在存在xUxxfoxx2.有界性定理(保序性).),()(),,(,0.)(lim,)(lim000BAxgxfxUxBxgAxfoxxxx则有若设3.不等式性质).0(0),0)((0)(,),(,0,)(lim000AAxfxfxUxAxfxx或则或时当且若推论).0)((0)(,),(,0),0(0,)(lim00xfxfxUxAAAxfoxx或时当则或且若定理(保号性)证明：)),,((||0,0,200xUxxxAo即取.23)(20,2|)(|AxfAAAxf推论).()(),,(,0,)(lim,)(lim000xgxfxUxBABxgAxfoxxxx有则且设六、极限运算法则⒈四则运算法则定理.0,)()(lim)3(;)]()(lim[)2(;)]()(lim[)1(,)(lim,)(limBBAxgxfBAxgxfBAxgxfBxgAxf其中则设证明：)(lim)(lim)()(lim000xgxfxgxfxxxxxx（利用Heine定理也可证）,0|||)(|lim,0)(lim00BxgBxgxxxx设.2|||)(|Bxg时当),(,0101xUxo时，),(,0,)(lim2020xUxAxfoxx,|)(|Axf|)(||||)()(||)()(|xgBxAgxBfBAxgxf22|)||(|2)|||(|2BBAABB|))(||))(||(|||22BxgAAxfBB时当取),(,,,min0321xUxo,|)(|),(,0,)(lim3030BxgxUxBxgoxx时，⒉复合运算)]([)(tgftgf定理,)(lim,)(lim000xtgAxfttxx设.)(lim)(lim00Axftgfxxtt则,)()(00xtgtUo内且在解释：相当于计算中进行变量替换，xtg)(证明：.|)]([|Atgf,)(lim0Axfxx,0,0,),(0时当xUxo.|)(|Axf,0对上述,0,),(0时当tUto,|)(|0xtg),,()(0xUtgo⑴例8、11lim321xxx)1)(1()1)(1(lim21xxxxxx32⑵⑶49lim22xxxxx22411921limxxxxx1023lim3xxx⑷232732lim2732lim42025230xxxxxxxxxxxx⑸xxnx1)1(lim011lim1uununuuunnu)1(lim211ux1AC七、两个重要极限(1)1sinlim0xxx)20(,,xxAOBO圆心角设单位圆,tan,,sinACxABxBDx弧于是有xoBD.ACO，得作单位圆的切线,xOAB的圆心角为扇形,BDOAB的高为,tansinxxx,1sincosxxx即.02也成立上式对于x,20时当xxcos12sin22x2)2(2x,22x,02lim20xx,0)cos1(lim0xx,1coslim0xx,11lim0x又.1sinlim0xxx0例9⑴xxxtanlim0),1coslim(0由刚才过程可知xx⑵的应用1sinlim0xxxxxxxcos1sinlim01bxaxxsinsinlim0babxbxaxaxxsinsinlim0ba⑶20cos1limxxx220)2(sin2limxxx4)2()2(sinlim2220xxx21)sin(lim2120ttt⑷xxxarcsinlim0txsin1sinlim0tttxxxarctanlim0txarctantxtan1tanlim0ttt2tan)1(lim1xxxtx1tx1)22tan(lim0ttttttt2cos2sinlim0⑸⑹ttt2cotlim022sin2lim20ttt(2)exxx)11(lim})11{(}{nnnx曾考虑数列.lim存在nnxennn)11(lim记为)71828.2(e,1时当x,1][][xxx有xx)11()][11(lim)][11(lim)][11(lim][1][xxxxxxxx而,e11][][)1][11(lim)1][11(lim)1][11(limxxxxxxxx,e.)11(limexxx][)1][11(xx,)][11(1][xx,xt令ttxxtx)11(lim)11(limttt)111(lim)111()111(lim1tttt.eexxx)11(lim,1xt令ttxxtx)11(lim)1(lim10.eexxx10)1(lim例10.xxx)21(lim极限的应用1tx2tx22)2(10)1(limettt直接凑222])21[(limexxx⑴211])11[()11(lim1111limeeexxxxxxxxx⑵思考题2.求极限xxxx193lim解答xxxx193limxxxxx111319limxxxxx313311lim9990e作业(数学分析习题集)习题2.3函数的极限A6(2),(3),(6),(7);7(1),(2)；9；10；11.习题2.4极限过程的其他形式及其性质A5；6；7(1),(4),(6)；8(1),(4)；11；12；14.