1 / 12

Mastering Limits in Calculus: Infinity and Asymptotes Mastery

Understand limits involving infinity, vertical and horizontal asymptotes in calculus. Learn definitions, examples, and computational methods. Find horizontal and vertical asymptotes for given curves easily.

mburchett
Download Presentation

Mastering Limits in Calculus: Infinity and Asymptotes Mastery

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Math 1304 Calculus I 2.6 – Limits involving Infinity, Asymptotes

  2. Recall: Limits of Infinity

  3. Recall: Vertical Asymptotes • Definition: The line x = a is called a vertical asymptote of the curve y = f(x) if any of the following limits exist

  4. Limits as x approaches ±infinity Now consider:

  5. Basic Examples • f(x) = 1/x • f(x) = 1/(x-a) • f(x) = 1/x2 • f(x)= 1/(x-a)2

  6. Def of Limit as x approaches infinity • Definition: Let f be a function defined on some interval (a,). We say that the limit as x approaches infinity is L if for any >0 there is a number N such that |f(x)-L|<  whenever x > N. In this case we write

  7. Def of Limit as x approaches -infinity • Definition: Let f be a function defined on some interval (-,a). We say that the limit as x approaches minus infinity is L if for any >0 there is a number N such that |f(x)-L|<  whenever x < N. In this case we write

  8. Def of Limit of infinity as x approaches infinity • Definition: Let f be a function defined on some interval (a,). We say that the limit as x approaches infinity is infinity if for any M there is a number N(M) such that f(x)>M whenever x > N(M). In this case we write

  9. Horizontal Asymptotes • Definition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either of the following limits exist

  10. Computational Methodsfor limits as x approaches +-infinity • Rules • Basic functions: constants, 1/x, 1/xr • Sum, difference, product, quotient, power, etc. • Algebraic Techniques • For quotient of polynomials, divide by highest power (example next)

  11. Basic Functions

  12. Examples • Find all horizontal and vertical asymptotes of the curve y=(2x-1)/(5x+3) • Find all horizontal and vertical asymptotes of the curve y=(2x2-1)/(3x2+3x+6)

More Related