导函数的极限与函数渐近线的关系

Relationships between the Limits of Derivative Functions and the Asymptotes of Functions

导函数作为一类非常重要的函数,其自身具有一些独特的性质,在研究函数过程中起到了重要的作用。函数的渐近线同样也是函数性质的一个重要表现。导函数是否具有水平渐近线刻划的就是导函数在在自变量趋向无穷大时是否有极限的问题。通过对函数及导函数极限的讨论,获得函数具有水平渐近线和导函数具有水平渐近线的关系,以及函数,导函数,二阶导数和三阶导数具有水平渐近线的条件,此外,对已有的一个研究结果进行了改进。

The derivative function is one of very important function, it not only has some unique properties, but also plays a very important role in studying functions. The asymptote of the function is also an important expression of the functional properties. Whether the derivative function has a horizontal asymptote is equivalent to the question that whether the derivative function has a limit when the independent variable tends to infinity. Through discussing the limits of the function and the derivative function, we obtain the relationship between the function has horizontal asymptote and the derivative function has horizontal asymptote, moreover, we obtain the conditions of the function, the derivative, the second derivative and the third derivative having the horizontal asymptote. In addition, one of the existing research results is improved.