函数列积分极限问题的多种解法与比较分析

Multiple Solutions and Comparative Analysis of the Limit Problem of Integral of Function Sequence

摘要: 本文以一道典型的函数列积分极限试题为研究对象，系统给出六种求解方法，分别基于夹逼准则、换元变形、分部积分递推、积分第一与第二中值定理、柯西–施瓦茨不等式及勒贝格控制收敛定理。通过对各方法的思路解析、知识点梳理与思维层次分析，展现一题多解在深化微积分理解、拓展解题思路中的作用。进一步探讨了方法的一般化推广与变式问题求解，并通过多维度对比表格进行系统比较，为函数列积分极限的学习与教学提供参考。

Abstract: Taking a typical test question of the limit of integral of function sequence as the research object, this paper systematically presents six solving methods, which are respectively based on the squeeze theorem, substitution transformation, integration by parts recursion, the first and second mean value theorems for integrals, Cauchy-Schwarz inequality and Lebesgue dominated convergence theorem. Through the analysis of ideas, knowledge points and thinking levels of each method, this paper demonstrates the role of multiple solutions to one problem in deepening the understanding of calculus and expanding problem-solving ideas. It further discusses the generalization of methods and the solution of variant problems, and conducts a systematic comparison through a multi-dimensional comparison table, providing reference for the learning and teaching of the limit of integral of function sequence.