Aim: Do Limits at Infinity make sense?

1. Aim: Do Limits at Infinity make sense? Do Now: List the characteristics of the following function

2. y = 3 x decreases w/o bound x increases w/o bound f(x) approaches 3 f(x) approaches 3 Do Now f(x)  3 as x  - f(x)  3 as x  

3. The line y = L is a horizontal asymptote of the graph of f if or y = 3 Horizontal Asymptote f(x)  3 as x  - f(x)  3 as x  

4. Limits at Infinity for Rational Functions For the rational function f(x) = P(x)/Q(x), where P(x) = anxn + . . . .a0 and Q(x) = bmxm + . . . .b0 the limit as x approaches positive or negative infinity is as follows: If n > m, the limit does not exist. n is the highest power of numerator; m is highest power of denominator

5. If x < 0, then Definition of Limits at Infinity If f is a function and L1 and L2 are real numbers, the statements Limit as x approaches - Limit as x approaches  denote the limits at infinity. The first is read “the limit of f(x) as x approaches - is L1” and the second is read “the limit of f(x) as x approaches  is L2”. If n is a positive real number, then

6. Find the limit: Operations with Limits If r is a positive rational number and c is any real number, then Also, if xr is defined when x < 0, then 5 – 0 = 5

7. graphically from study of rational polynomial functions If degree of p = degree of q, then the line y = an/bm is a horizontal asymptote, the ratio of the coefficients. Evaluating Limit at Infinity Find the limit: Note: a rational function will always approach the same horizontal asymptote to the right and left.

8. horizontal asymptote y = 1/2 x approaches  x approaches - vertical asymptote x = 0 Limits at Infinity What happens to f(x) as x approaches  infinity?

9. Algebraically: Model Problem Find the limit The limit of f(x) as x approaches  is 4. Graphically:

10. Application You are manufacturing a product that costs $0.50 per unit to produce. Your initial investment is $5000, which implies that the total cost of producing x units is C = 0.5x + 5000. The average cost per unit is Find the average cost per unit when a) x = 1000, b) x = 10,000, c) when x 

11. = 2 Evaluating Limit at Infinity Find the limit: algebraically: divide all terms by highest power of x in denominator

12. Model Problem Find the limit for each as x approaches . A polynomial tends to behave as its highest-powered term behaves as x approaches   no limit

13. Functions with Two Horizontal Asymptotes Determine limits for each

14. Functions with Two Horizontal Asymptotes Determine limits for each

15. Limits Involving Trig Functions Determine limits for each as x approaches infinity x oscillates between -1 and 1 conclusion: a limit does not exist f(x) oscillates between two fixed values as x approaches c.

16. because -1 < sin x < 1, for x > 0, Limits Involving Trig Functions Determine limits for each Squeeze Theorem:

17. Model Problem 1 f(t) measures the level of oxygen in a pond, where f(t) = 1 is the normal (unpolluted) level and the time t is measured in weeks. When t = 0, organic waste is dumped into the pond, and as the waste material oxidizes, the level of oxygen in the pond is What percent of the normal level of oxygen exists in the pond after 1 week? After 2 weeks? After 10 weeks? What is the limit as t approaches infinity?

18. Model Problem 1 after 1 week? After 2 weeks? After 10 weeks? What is the limit as t approaches infinity? 50% 60% 90.1%

19. Model Problem 1 What is the limit as t approaches infinity? divide all terms by highest power of t in denominator

20. Limits at Infinity • Let L be a real number. • The statement means that for each  > 0 there exists an M > 0 such that whenever x > M. • The statement means that for each  > 0 there exists an N < 0 such that whenever x < N.

21. f(x) is within  units of L as x    L  M Limits at Infinity For a given positive number  there exists a positive number M such that, for x > M, the graph of f will lie between the horizontal lines given by y = L +  and y = L - .