1. 1.5 Infinite Limits Chapter 1 – Larson- revised 9/12

2. Blast from the Past Find the limit (if it exists) 1. 2. 3. 4.

3. Blast from the Past Find the limit (if it exists) 5. 6. where 7. Determine the value of c such that the function is continuous on the entire real line.

4. Blast from the Past Find the limit (if it exists) 1. 2. 3. 4.

5. Blast from the Past Find the limit (if it exists) 5. 6. 7. Determine the value of c such that the function is continuous on the entire real line. c =

6. 1. 5 Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative. vertical asymptote at x =0.

7. Vertical Asymptote Slide 2- 7 Determine all vertical asymptotes of the graph:

8. Vertical Asymptote Slide 2- 8 First simplify the expression Therefore is a vertical asymptote, however note that is not.

9. Examples : The denominator is positive in both cases, so the limit is the same. Try: 1. 2. 3.

10. Examples : 1. 2. 3.

11. Properties of Limits: Where f(x) and g(x) are functions with limits that exists such that and 1.Sum or difference: 2.Product: 3.Quotient:

12. As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or 3.5 Limits at Infinity

13. Horizontal Asymptote Slide 2- 13 For example the following limit is: Therefore the graph of the function has a horizontal asymptote at

14. Limits at Infinity If r is a positive rational number and c is any real number, then Furthermore, if is defined when, then

15. End Behavior Models Right-end behavior models give us: dominant terms in numerator and denominator

16. Comparison of Three Rational Functions a. b. c.

17. A Function with Two Horizontal Asymptotes This number becomes insignificant as. There is a horizontal asymptote at 1. There is a horizontal asymptote at -1

18. Often you can just “think through” limits.  Numerator grows faster Denominator grows faster