文档介绍：上页下页结束返回首页铃一、准则I 及第一个重要极限二、准则II 及第二个重要极限§ 两个重要极限上页下页铃结束返回首页上页下页铃结束返回首页一、准则I及第一个重要极限如果数列{xn}、{yn}及{zn}满足下列条件?(1)yn?xn?zn(n?1? 2? 3????)??准则I (夹逼定理)?准则I ?如果函数f(x)、g(x)及h(x)满足下列条件?(1) g(x)?f(x)?h(x)? (2)lim g(x)?A?limh(x)?A?那么limf(x)存在?且limf(x)?A?(2)aynn???lim?aznn???lim?下页那么数列{xn}的极限存在?且??nlimxn?a?那么数列{xn}的极限存在?且??nlimxn?a?上页下页铃结束返回首页?第一个重要极限显然BC?AB?AD?(因此sin x?x? tan x ?DB1OCAx1sinlim0??xxx?简要证明?参看附图?设圆心角?AOB?x(20???x)?从而1sincos??xxx (此不等式当x?0时也成立)?因为1coslim0??xx?根据准则I??1sinlim0??xxx?下页上页下页铃结束返回首页?应注意的问题这是因为?令u??(x)?则u?0?于是?第一个重要极限1sinlim0??xxx?在极限)()(sinlimxx??中?只要?(x)是无穷小?就有1)()(sinlim?xx???)()(sinlimxx??1sinlim0???uuu?下页上页下页铃结束返回首页1sinlim0??xxx?1)()(sinlim?xx??(?(x)?0)?例1?例1?求xxxtanlim0??解?xxxtanlim0?xxxxcos1sinlim0???1cos1limsinlim00?????xxxxx?解?解?xxxtanlim0?xxxxcos1sinlim0???1cos1limsinlim00?????xxxxx?解?xxxtanlim0?xxxxcos1sinlim0???1cos1limsinlim00?????xxxxx?解?xxxtanlim0?xxxxcos1sinlim0???1cos1limsinlim00?????xxxxx?解?例2?例2?求20cos1limxxx???解?20cos1limxxx???220220)21(2sinlim212sin2limxxxxx???2112122sinlim21220???????????????xxx?解?20cos1limxxx???220220)21(2sinlim212sin2limxxxxx???2112122sinlim21220???????????????xxx?下页220sin12lim22xxx??? ?? ?? ?上页下页铃结束返回首页1sinlim0??xxx?1)()(sinlim?xx??(?(x)?0)?例3?解?解?例4?)0,0.(sinsinlim0???nmnxmxx求nxnxnxmxmxmxnxmxxxsinsinlimsinsinlim00?????.nm?.sintanlim30xxxx??求2030cos1tanlimsintanlimxxxxxxxxx??????200cos1limtanlimxxxxxx?????21?下页上页下页铃结束返回首页1sinlim0??xxx?1)()(sinlim?xx??(?(x)?0)?例5?解?.arcsinlim0xxx?求,arcsinxy?令,sinyx?则,0,0??yx时当yyxxyxsinlimarcsinlim00???1?下页上页下页铃结束返回首页1sinlim0??xxx?1)()(sinlim?xx??(?(x)?0)?例6?解???求,tx???令,0,??tx时当?53??)55tan()33sin(lim5tan3sinlim0ttxxtx????????ttt5tan3sinlim0???535tan533sinlim0ttttt???上页下页铃结束返回首页二、准则II及第二个重要极限?单调数列如果数列{xn}满足条件x1?x2?x3?????xn?xn?1?????就称数列{xn}是单调增加的?如果数列{xn}满足条件x1?x2?x3?????xn?xn?1?????就称数列{xn}是单调减少的?单调增加和单调减少数列统称为单调数列?下页上页下页铃结束返回首页?准则II单调有界数列必有极限?前面曾证明?收敛的数列一定有界?但有界的数列不一定收敛?现在准则II表明?如果数列不仅有界?并且是单调的?那么这个数列一定是收敛的??说明下页