第1章函數(shù)與極限習(xí)題解答

1、第1章函數(shù)與極限習(xí)題解答函數(shù)與極限習(xí)題解答1第1章函數(shù)與極限習(xí)題解答函數(shù)與極限習(xí)題解答1.兩個(gè)無(wú)窮小的商是否一定是無(wú)窮??？舉例說(shuō)明之.解不一定.例如當(dāng)x?0時(shí)?(x)?2x?(x)?3x都是無(wú)窮小但不是32)()(lim0??xxx??)()(xx??無(wú)窮小.2.函數(shù)y?xcosx在(????)內(nèi)是否有界？這個(gè)函數(shù)是否為當(dāng)x???時(shí)的無(wú)窮大？為什么？解函數(shù)y?xcosx在(????)內(nèi)無(wú)界.這是因?yàn)?M?0在(????)內(nèi)總能找到這樣的

2、x使得|y(x)|?M.例如y(2k?)?2k?cos2k??2k?(k?012???)當(dāng)k充分大時(shí)就有|y(2k?)|?M.當(dāng)x???時(shí)函數(shù)y?xcosx不是無(wú)窮大.這是因?yàn)?M?0找不到這樣一個(gè)時(shí)刻N(yùn)使對(duì)一切大于N的x都有|y(x)|?M.例如(k?012???)0)22cos()22()22(???????????kkky對(duì)任何大的N當(dāng)k充分大時(shí)總有但|y(x)|?0?M.Nkx???22??3.證明:函數(shù)在區(qū)間(01]上無(wú)界但這

3、函數(shù)不是當(dāng)x?0時(shí)的無(wú)窮大.xxy1sin1?證明證明函數(shù)在區(qū)間(01]上無(wú)界.這是因?yàn)閤xy1sin1??M?0在(01]中總可以找到點(diǎn)xk使y(xk)?M.例如當(dāng)(k?012???)221????kxk時(shí)有22)(????kxyk當(dāng)k充分大時(shí)y(xk)?M.當(dāng)x?0時(shí)函數(shù)不是無(wú)窮大.這是因?yàn)閤xy1sin1??M?0對(duì)所有的??0總可以找到這樣的點(diǎn)xk使0?xk??但y(xk)?M.例如可取(k?012???)?kxk21?當(dāng)k充分

4、大時(shí)xk??但y(xk)?2k?sin2k??0?M.4.計(jì)算下列極限計(jì)算下列極限:(1)121lim22?????xxxx第1章函數(shù)與極限習(xí)題解答函數(shù)與極限習(xí)題解答3(9)xxx10)21(lim??解.??22210221010)21(lim)21(lim)21(limexxxxxxxxx??????????(10)xxxx2)1(lim???解.??222)11(lim)1(limexxxxxxx????????5.利用極限存在準(zhǔn)

5、則證明利用極限存在準(zhǔn)則證明:(1)111lim????nn證明證明因?yàn)槎矣蓸O限存在準(zhǔn)則I.nn11111????11lim???n1)11(lim????nn111lim????nn(2)??11211lim222???????????????nnnnnn證明證明因?yàn)?????????????????????22222221211nnnnnnnnnn而1lim22?????nnnn1lim22?????nnn所以???11211lim

6、222???????????????nnnnnn(3).??11lim0???xxx證明證明因?yàn)樗?又因?yàn)楦鶕?jù)夾逼準(zhǔn)則??xxx1111?????111???xxx11lim)1(lim00???????xxx有.??11lim0???xxx6.無(wú)窮小概念題無(wú)窮小概念題(1)當(dāng)x?0時(shí)?2x?x2與x2?x3相比?哪一個(gè)是高階無(wú)窮?。拷庖?yàn)?2lim2lim202320????????xxxxxxxxx所以當(dāng)x?0時(shí)?x2?x3是高階

7、無(wú)窮小即x2?x3?o(2x?x2).(2)當(dāng)x?1時(shí)?無(wú)窮小1?x和(ⅰ)1?x3(ⅱ)是否同階？是否等價(jià)？)1(212x?解(ⅰ)因?yàn)?)1(lim1)1)(1(lim11lim212131??????????????xxxxxxxxxxx所以當(dāng)x?1時(shí)1?x和1?x3是同階的無(wú)窮小但不是等價(jià)無(wú)窮小.(ⅱ)因?yàn)?)1(lim211)1(21lim121???????xxxxx所以當(dāng)x?1時(shí)1?x和是同階的無(wú)窮小而且是等價(jià)無(wú)窮小.)1