基于积分定义求解极限问题的方法综述

A Survey of Limit-Solving Methods via Integral Definitions

摘要: 本文针对积分定义计算特殊极限问题展开：首先，给出黎曼和与定积分、二重积分和三重积分定义之间的理论联系，再给出黎曼和与积分的转化方法；其次，通过等分、特殊节点的选取，将区间或者区域分割，给出将极限转化为定积分、二重积分和三重积分的具体方法；最后，文中主要选取历年数学竞赛和考研试题，对所提方法加以实践和应用。

Abstract: This paper addresses the problem of evaluating special limits via integral definitions. First, the theoretical connection between Riemann sums and the definitions of definite, double, and triple integrals is established, and the transformation methodology from Riemann sums to integrals is formulated. Second, through equipartition and specific sample-point partitioning of intervals or regions, concrete methods for converting limits into definite, double, and triple integrals are derived. Finally, the proposed methods are applied and validated through problems selected from previous mathematical competitions and national graduate entrance examinations.