2講極限

1、第二講第二講極限極限一、數(shù)列極限一、數(shù)列極限例1（1）（2）1lim0nn???212lim525xnn?????（3）0.10.110.1110.111的極限為???19（4）=？112332lim??????nnnnn二、函數(shù)極限二、函數(shù)極限例2（1）；（2）；（3）當時??2lim324xx???21214lim221xxx?????00?x00limxxxx??三、三、左右極限與極限的關(guān)系左右極限與極限的關(guān)系定理：定理：的充分必

2、要條件是:??Axfxx??0lim????.limlim0000Axfxfxxxx??????因此即使和都存在但若它們不相等則仍不存在??xfxx00lim????xfxx00lim????xfxx0lim?例3已知證明不存在.??100010.xxfxxxx?????????????xfx0lim?證因為即????11limlim00????????xxfxx????11limlim00???????xxfxx????xfxfxx?

3、????00limlim故不存在(如圖所示).??xfx0lim?例4設(shè)討論時函數(shù)的極限的存在性??0sin0.xxfxxx??????0?x??xf[注]時的左、右極限常用下列記號:0?x??????????????????????????0000limlimlim0000fffxfxfxfxxx??????????????????????????0000limlimlim0000fffxfxfxfxxx(2)已知為常數(shù)則ba12li

4、m2??????????????baxxxx?a?b(3)已知為常數(shù)則.ba21lim1????xbaxx?a?b2.求下列極限:（1）??nnnn????1lim（2）nnn31313112121211lim22????????????（3）??1lim21??????xnxxxnx?(4)(為正整數(shù))11lim1???mnxxxnm3.設(shè)和為非負整數(shù)均為常數(shù)求0000??bamn????mjbniaji210210????極限:10