极限的多种求法.doc

目 录 摘 要 1 前 言 2 1极限的定义 3 1.1数列极限的概念 3 1.2 函数极限的概念 3 ⑴趋于时函数的极限 3 ⑵趋于时函数的极限 4 ⑶趋于时函数的极限（函数极限的定义） 4 ⑷趋于时的左右极限 4 2数列极限的求法 5 2.1利用数列极限定义证明极限成立。 5 2.2利用迫敛性求极限 5 2.3利用数列极限四则运算法则求极限 7 2.4利用无穷小量与无穷大量互为倒数的关系求极限 8 2.5利用单调有界必有极限定理求数列极限 9 2.6利用级数收敛的必要条件求数列极限 10 2.7 利用重要极限求极限 10 2.8 利用定积分求极限 11 3函数极限的求法 12 3.1用函数极限定义证明极限成立 12 3.2利用左右极限与函数关系求极限（适用于分段函数求分段点处的极限等情况） 13 3.3利用函数连续性求极限（适用于求函数在连续点处的极限） 13 3.4利用两个重要极限求极限 14 3.5用函数四则运算法则求极限 15 ⑴用未知数的最高次方项去除分子、分母 16 ⑵分子有理化 16 ⑶分母有理数 16 ⑷三角函数 16 3.6 初等变形求极限 17 3.7 利用“有界函数与无穷小之积仍为无穷小”之性质求极限 17 3.8 利用无穷大量与无穷小量互为倒数的关系求极限 18 3.9利用等价无穷小代替求极限 19 3.10 代数函数的极限 20 3.11利用洛必达法则求极限（适用于未定式极限） 21 (1) 型不定式 22 (2) 型不定式 22 ⑶其它不定式 22 3.12利用拉格朗日中值定理求极限 24 3.13利用泰勒公式 25 参考文献 26 摘 要 摘要Abstract: limit plays an important role in mathematical analysis, this paper introduces some methods for the limit. Methods this paper finished mainly the following: one, the limit of number sequence for 1 definition 2 forced convergence method 3 four algorithms for limit 4 using infinitesimal and infinity is the reciprocal relationship limit 5 monotone bounded will limit theorem of limit of a sequence of 6 using the necessary conditions for convergence of series the limit of a sequence of 7 important use limit limit. Method two, function limit: 1 Definitions of function limit prove limit was founded in 2 by left and right limit and function limit (suitable for piecewise function for the piecewise point limit etc.) 3 using the continuity of function limit (suitable for the function in the continuous point limit) 4 using two important limits for the limit of 5 6 primary deformation limit 7 use bounded function and an infinitesimal is an infinitesimal product of nature limit 8 using the infinity and infinitesimal reciprocal relations for limit 9 utilization of infinitesimal instead of price limit of 10 algebraic function limit function using four algorithms for limit. And the use of LHospital Rule limit (applicable to the limits), using the Lagrange mean value theo