2025周洋鑫考研数学全程班零基础提前学作业新浪微博@考研数学周洋鑫12025年考研数学零基础提前学同步作业作业4·七种未定式极限专题计算【1】【解析】方法一：利用泰勒公式.()()()()333300333311arcsinsin66limlim11arctantan33xxxxoxxxoxxxxxxxoxxxox→→++−−+−=−−+−++()()3330033311133limlim22233xxxoxxxoxx→→+===−−+−.方法二：巧妙创造等价无穷小.()()()()00arcsinsinarcsinsinlimlimarctantanarctantanxxxxxxxxxxxxxx→→−+−−=−−+−330331166lim1133xxxxx→+=−+−（符合加减法等价无穷小的替换准则）303113lim223xxx→==−−.【2】【解析】()2arctantanarctantan300limlim1sinxxxxxxxeeeexex→→−−=−()tanarctantan301limxxxxeex−→−=30arctantanlimxxxx→−=()()3333301133limxxxoxxxoxx→−+−++=()3330223lim3xxoxx→−+==−.2025考研数学全程班零基础提前学作业新浪微博@考研数学周洋鑫2【3】【解析】()0ln1lncoslimsinxxexxxx→+−03lncoslim16xxxx→=（等价无穷小替换、非零因子先算）02lncoslim16xxx→=02cos1lim16xxx→−=20212lim316xxx→−==−.【4】【解析】01sin1limtanxxxxx→+−+−()()0sinlimtan1sin1xxxxxxx→−=−+++（分子有理化）01sinlim2tanxxxxx→−=−（非零因子淡化）3031116lim1243xxx→−==−.【5】【解析】021sincos2limtan2xxxxx→+−021sincos2limtan(1sincos2)2xxxxxxxx→+−=++（分子有理化）0211sincos2lim2tan2xxxxx→+−=（非零因子淡化）20sin(2sin)lim22xxxxx→+=（二倍角公式化简）0001112222sin2sinlimlimlim246xxxxxxxxxx→→→+==+=+=.【6】【解析】201e1arctanlim1cosxxxxx+→−+−2025考研数学全程班零基础提前学作业新浪微博@考研数学周洋鑫3201e1arctanlim12xxxxx+→−+=（等价无穷小代换）2001arctane1limlim1122xxxxxxx++→→−=+（四则运算法则）001limlimarctan212xxxxxx++→→=+（1arctanx为有界函数）202=+=．【7】【解析】()()3224ln21lnlimlimlnln54xxxxxxexxeeexx→+→++++=+++（抓大头）1lim22xxx→+==.【8】【解析】令xt=−，则()22lim11xxxxx→−+−−−+()22lim11ttttt→+=−−−++2222lim11tttttt→+−−=−−+++（有理化）222limtttt→+−=+（抓大头）1=−.【9】【解析】222limtanlimsin2cosxxxxxxx→→−−=...
