Calculus1-Lecture 24|Asymptotes, Vertical Tangents and Cusps Explained
Автор: Shadi M. Shaqaqha
Загружено: 2026-03-14
Просмотров: 8
Описание:
In this lecture, we study vertical and horizontal asymptotes and analyze the behavior of functions near these asymptotes.
Topics covered:
Definition of vertical asymptotes
Definition of horizontal asymptotes
Using limits to determine asymptotes
Sign analysis near vertical asymptotes
Examples involving rational and trigonometric functions
Vertical tangents
Vertical cusps and how they differ from vertical tangents
Several examples illustrate how limits and derivatives describe the behavior of graphs near asymptotes and special points.
Instructor: Prof. Shadi Shaqaqha
Department of Mathematics – Yarmouk University
شرح المقاربـات الرأسية والأفقية (Asymptotes) وكذلك المماس الرأسي والنقطة الحادة (Vertical Cusp) مع تطبيقات على دوال كسرية ومثلثية.
00:00 Vertical asymptotes – definition
01:58 Horizontal asymptotes – definition
02:47 Example – asymptotes of 2x/(x²−9)
07:49 Example – asymptotes of cos(x)/x
12:10 Example – asymptotes of (x/(1−2x))³
13:41 Example – asymptotes of √x /(2√x − x − 1)
18:45 Vertical tangents – definition
19:49 Vertical cusps – definition
20:47 Examples – tangent vs cusp classification
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