兩個重要極限

1、11.6兩個重要極限兩個重要極限準則準則I如果數(shù)列xn、yn及zn滿足下列條件?(1)yn?xn?zn(n?1?2?3????)?(2)??aynn???limaznn???lim那么數(shù)列xn的極限存在?且?axnn???lim證明?因為??以根據(jù)數(shù)列極限的定義????0??Naynn???limaznn???lim1?0?當n?N1時?有|yn?a|???又?N2?0?當n?N2時?有|zn?a|???現(xiàn)取N?maxN1?N2?則當n

2、?N時?有|yn?a|???|zn?a|???同時成立?即a???yn?a???a???zn?a???同時成立?又因yn?xn?zn?所以當n?N時?有a???yn?xn?zn?a???即|xn?a|???這就證明了?axnn???lim簡要證明?由條件(2)????0??N?0?當n?N時?有|yn?a|??及|zn?a|???即有a???yn?a???a???zn?a???由條件(1)?有3?xxxcos1sin1??或?1sinc

3、os??xxx注意此不等式當??x?0時也成立?而?根據(jù)準則I??2?1coslim0??xx?1sinlim0??xxx簡要證明簡要證明?參看附圖?設圓心角?AOB?x()?20???x顯然BC?AB?AD?因此sinx?x?tanx?從而(此不等式當x?0時也成立)?1sincos??xxx因為?根據(jù)準則I???1coslim0??xx1sinlim0??xxx應注意的問題?在極限中?只要?(x)是無窮小?就有?)()(sinlim

4、xx??1)()(sinlim?xx??這是因為?令u??(x)?則u?0?于是?)()(sinlimxx??1sinlim0???uuu?(?(x)?0)?1sinlim0??xxx1)()(sinlim?xx??例1?求?xxxtanlim0?解??xxxtanlim0?xxxxcos1sinlim0???1cos1limsinlim00?????xxxxx例2?求?20cos1limxxx??解??20cos1limxxx??22