高等數(shù)學-極限與連續(xù)(習題)

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

2、xn1ln?2、設問9.01?u99.02?u?????個nnu999.0?(1)lim???nnu(2)應為何值時才能使與其極限之差的絕對值小于？nnu0001.0解：(1)顯然可見；nnu1011??1lim???nnu(2)欲使只需即可.41010001.0101|1|????nnu5?n那么當時有無窮多個滿足0???1??NNn?nx???||axn但顯然不存在.nnx??lim(4)設數(shù)列若對中僅有有限個不滿足則nx0???n

3、xnx???||axn.axnn???lim解：結(jié)論正確.假設僅有不滿足0???knnnxxx21????||axn于是取那么當時???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