1. 2.2 Limits Involving Infinity

2. The symbol  The symbol  means unbounded in the positive direction. (-  in negative direction) It is NOT a number!

3. As the denominator gets larger, the value of the fraction gets smaller. In other words as x gets larger positively or negatively, the y-values get closer to zero. The line y = b is a horizontal asymptote if: or The line y = 0 is a horizontal asymptote for f

4. As the denominator approaches zero from the left, the value of the fraction gets very large. vertical asymptote at x =0. As the denominator approaches zero from the right, the value of the fraction gets very large negatively.

5. Review: Finding Asymptotes 1 st make sure R(x) = p(x)/q(x) is in simplest terms VerticalHorizontalOblique Deg top > deg bottom Set bottom = to 0 and solve for x noneDivide top by bottom, y=answer (no remainder) Deg top = deg bottom Set bottom = to 0 and solve for x y = quotient of leading coeff of top and bottom none Deg top < deg bottom Set bottom = to 0 and solve for x y = 0none

6. Examples: Find asymptotes and graph

7. Vertical Asymptotes- Infinite Limits The vertical line x = a is a vertical asymptote of a function y = f(x) if If

8. Graphically

9. Examples: Find the limits graphically and numerically

11. Horizontal Asymptotes – Limits at Infinity The line y = b is a horizontal asymptote of y = f(x) if either The limit at infinity is also referred to as end behavior.

12. Examples: Find the limits at infinity graphically and numerically

13. Finding the limit at infinity analytically If f(x) is a rational function then to find the limit at infinity simply find the horizontal asymptote using the rules about degrees.

14. Examples

15. Theorem

16. Non-rational functions If the function is not a rational function then you can try: 1.Dividing top and bottom by highest power on bottom 2.Rationalizing 3.Rewriting the problem

17. Examples: Divide

18. Example: Rationalize

19. Example: Rewrite

20. End Behavior Models Graph on the window [-20, 20] by [-1000000, 5000000] Notice as the graphs become identical. We say that g(x) act as a model for f(x) as or g(x) is an end behavior model for f(x)

21. Example Show graphically that g(x) = x is a right end behavior model and h(x) = e -x is a left end behavior model for f(x) = x + e -x

22. End behavior models for polynomials If

23. Examples: Find the end behavior model

24. HW: p. 71 1-22,29-38 Worksheet