高等數(shù)學(xué)-極限與連續(xù)(習(xí)題)

1、第二章第二章極限與連續(xù)極限與連續(xù)習(xí)題習(xí)題21211、觀察下列數(shù)列的變化趨勢(shì)判別哪些數(shù)列有極限如有極限寫(xiě)出它們的極限.(1)；有..nnax1?)1(?a0lim???nnx(2)；有..nxnn1)1(1???0lim???nnx(3)；無(wú).nxnn1)1(???(4)；無(wú).2sin?nxn?(5)；有..11???nnxn1lim???nnx(6)；無(wú).nnx)1(2??(7)；有..nxn1cos?1lim???nnx(8).無(wú).n

2、xn1ln?2、設(shè)問(wèn)9.01?u99.02?u?????個(gè)nnu999.0?(1)lim???nnu(2)應(yīng)為何值時(shí)才能使與其極限之差的絕對(duì)值小于？nnu0001.0解：(1)顯然可見(jiàn)；nnu1011??1lim???nnu(2)欲使只需即可.41010001.0101|1|????nnu5?n那么當(dāng)時(shí)有無(wú)窮多個(gè)滿足0???1??NNn?nx???||axn但顯然不存在.nnx??lim(4)設(shè)數(shù)列若對(duì)中僅有有限個(gè)不滿足則nx0???n

3、xnx???||axn.axnn???lim解：結(jié)論正確.假設(shè)僅有不滿足0???knnnxxx21????||axn于是取那么當(dāng)時(shí)???Nmax21knnnN?Nn????||axn所以.axnn???lim5、用極限性質(zhì)判別下列結(jié)論是否正確為什么？(1)若收斂則(為正整數(shù))；nxknnnnxx??????limlimk解：結(jié)論正確.顯然是的子數(shù)列故.knx?nxnnknnxx??????limlim(2)有界數(shù)列必收斂；nx解：結(jié)論錯(cuò)