渐近线问题研究

Research on Asymptotic Line

现今高等数学、数学分析教材中关于渐近线内容的研讨不尽完善,比较含糊,存在欠缺.为此,本文完善了渐近线的定义,依据本文给出的渐近线的定义求函数曲线的渐近线时,不会丢失渐近线.同时,本文对函数曲线与其渐近线的交点、函数的无穷间断点与函数曲线的垂直渐近线的关系、函数曲线的水平渐近线和斜渐近线的关系进行了系统的研讨.本文阐明了求函数曲线的渐近线的步骤和方法,指出不用斜渐近线的系数公式可以直接用斜渐近线的定义,将求斜渐近线的系数转化成求含有参数的极限.

discussion on asymptotic line can be found in advanced math material and math analysis textbooks,however,the discussions are not perfect,ambiguous and incomplete.Thus,the paper tries to improve the definition.The asymptotic line will not be lost if we use the definition given in this paper to find the asymptotic line of function curve.Meanwhile,systematic discussions are made on the following aspects - point of intersection between function curve and its asymptotic line,relation between infinite discontinuity of the function and vertical asymptote of the function curve,relation between horizontal asymptote of the function curve and oblique asymptote.The paper illustrates the steps and methods to find the asymptotic line of function curve.It also states that,by directly using definition of oblique as-ymptote other than coefficient formula of oblique asymptote,we can find limit with parameters other than coefficient of oblique asymptote.