%0 Journal Article
%T 利用极限与无穷小之间的关系快速求渐近线<br>Using the Relationship between Limit and In-finitesimal to Find Asymptote Quickly
%A 郭建立
%A 张曦丹
%A 晏建学
%J Advances in Applied Mathematics
%P 3753-3762
%@ 2324-8009
%D 2023
%I Hans Publishing
%R 10.12677/AAM.2023.128369
%X 函数图形描述的是“增减极值渐近线，凹凸拐点曲率圆”，其中渐近线描述函数图形变化趋势。求水平渐近线、斜渐近线需要针对函数关系y=f(x)分别考虑两个单侧极限(x→-∞或 <span>x→+∞</span>)，斜渐近线在第一次求出斜率之后还需要第二次求极限才能求出截距，垂直渐近线对应于函数的无穷间断点。隐函数F(x,y)=0由于难以得出函数关系y=f(x)，从而更加难以求出渐近线。本文梳理了显函数求垂直、水平、斜渐近线的四种题型及其快速解法，使得求渐近线快速简洁，同时给出了丰富的实例。创新之处在于利用极限与无穷小之间的关系快速简便求出渐近线，同时讨论了隐函数间接求垂直、水平、斜渐近线的方法。<br />
The Graph of a function describes “increasing or decreasing, extreme value, asymptote, concave, convex, inflection point, curvature circle”, in which the asymptote describes the change trend of the function graph. To calculate the horizontal Asymptote and the oblique Asymptote, two unilateral limits (<span>x→-∞</span> or <span>x→+∞</span> ) need to be considered respectively for the functional relationship <span>y=f(x)</span> . The oblique Asymptote needs to calculate the limit for the second time after calculating the slope for the first time to calculate the intercept. The vertical Asymptote corresponds to the in-finite breakpoint of the function. The Implicit function <span>F(x,y)=0</span> is more difficult to find the As-ymptote because it is difficult to find the functional relationship <span>y=f(x)</span> . This paper combs four types of problems and their fast solving process of explicit function to solve vertical, horizontal and oblique Asymptote, which makes solving Asymptote fast and concise, and gives a wealth of exam-ples. The innovation lies in using the relationship between the limit and the infinitesimal to quickly and simply find the Asymptote. At the same time, the method of indirectly finding vertical, hori-zontal and oblique Asymptote with Implicit function is discussed.
%K 显函数，隐函数，极限，无穷大，无穷小，无穷间断点，垂直渐近线，水平渐近线，
斜渐近线，四种题型<br>Explicit Function
%K Implicit Function
%K Limit
%K Infinity
%K Infinitesimal
%K Infinite Breakpoint
%K Vertical Asymptote
%K Horizontal Asymptote
%K Oblique Asymptote
%K Four Types of Problems
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=71169