一道全国大学生数学竞赛决赛题的推广

Generalizations for a Question of China Undergraduate Mathematical Contest in Finals

解答一道全国大学生数学竞赛非数学类决赛试题,该试题涉及微分方程,定积分及一元函数求极限.针对以积分形式表示的函数求极限问题,将定义在[0,1]区间上特定的被积函数分别推广到单调连续函数、连续函数及[-1,1]区间上的连续函数这三种形式.利用夹逼准则、连续函数的定义及反常积分一致收敛的性质可证推广命题成立.

A question of China undergraduate mathematical contest in finals for non- mathematics major is studied.The problem concerns the knowledge of differential equations,definite integral and limit of single variable function.For the limit of function,which is described by definite integral,the given integrand defined on the interval[0,1]is extended to the monotonic continuous function,continuous function on the interval [0,1]and continuous function on the interval[-1,1],respectively.Our proofs involve sandwich theorem,the definition of continuous function and the nature of improper integral of uniform convergence.