Limits and Continuity
description
Transcript of Limits and Continuity
![Page 1: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/1.jpg)
Limits and Continuity
Chapter 2:
![Page 2: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/2.jpg)
Aim #2.2 How do we solve limits as x approaches +/- infinity?
• We use infinity when the values in its domain or range outgrow all finite bounds.
• Graph y = 1/x • What happens as x approaches to + infinity?• What happens as x approaches to - infinity?
![Page 3: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/3.jpg)
Definition: Horizontal Asymptote
![Page 4: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/4.jpg)
Example 1: Looking for Horizontal Asymptotes
![Page 5: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/5.jpg)
Graph the following f(x)
![Page 6: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/6.jpg)
Theorem 5
![Page 7: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/7.jpg)
Example 5: Using the Theorem 5
![Page 8: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/8.jpg)
Infinite Limits as x ---a
• Graph the function f(x)= 1/x. What happens as x approaches 0 from the right?
• From the left? What does y approach?
![Page 9: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/9.jpg)
Definition: Vertical Asymptotes
![Page 10: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/10.jpg)
Example 4: Finding the Vertical Asymptote
What does f(x) approach as x approaches 0 from the right? From the left?
![Page 11: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/11.jpg)
Example 5: Finding Vertical Asymptotes
• What are the vertical asymptotes of • f(x)= tan x?
![Page 12: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/12.jpg)
End Behavior Models
• For numerically large values of x, we can sometimes model the behavior of a
complicated function by a simpler one that acts virtually the same way.
![Page 13: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/13.jpg)
Example 6:
![Page 14: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/14.jpg)
Definition End Behavior Models
![Page 15: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/15.jpg)
Example 7:
![Page 16: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/16.jpg)
Seeing Limits as x approaches +/- infinity
• We can investigate the graph of y =f(x) as x approaches +/- infinity by investigating the graph of y =f(1/x) as x approaches infinity.
![Page 17: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/17.jpg)
Example 9: Using Substitution
![Page 18: Limits and Continuity](https://reader035.fdocuments.us/reader035/viewer/2022081505/568166a5550346895dda9351/html5/thumbnails/18.jpg)
Summary: Answer in complete sentences.• Explain why there is no value L for which lim
as x approaches infinity for sin x = L