关于“当x→0时,sinx→0”结论的教材编写问题的探讨

Discussion on composing the conclusion "when x→0,sinx→0" in teaching material

针对同济大学应用数学系主编的《高等数学》教材中存在着关于"对结论‘当x→0时,sinx→0’未加证明的情况下而应用"的问题,提出三种解决的方法:第一种方法是在讲授无穷小概念之前、极限定义应用时,证明limx→0sinx=0;第二种方法是在讲授无穷小比较之前、第一类重要极限证明之后,证明limx→0sinx=0;第三种方法是把"无穷小比较"这一节放到本章最后,即在讲完函数连续性后学习。实践证明第二种是简单而实用的好方法。

Aiming at the problem ＂does not prove the conclusion‘when x→0 ,sinx→0＇ and applies it＂ existing in Advanced Mathematics composed by the Department of Applied Mathematics, Tongji University, puts forward three treatment methods. The first method, proves , when teaching limit definition application before infinitely small conception;the second method, proves after teaching the First Important Limit demonstration, before teaching infinitely small comparison ; the third method, puts the section of infinitely small comparison at the end of chapter, namely, learns it after teaching function continuity. Among them the second one is the most practical and simplest one.