1. Today: Limits Involving Infinity lim f(x) =  x -> a Infinite limits Limits at infinity lim f(x) = L x -> 

2. CHAPTER 2 2.4 Continuity Infinite Limits (see Sec 2.2, pp 98-101)

3. CHAPTER 2 2.4 Continuity Definition Let f be a function defined on both sides of a, except possibly at a itself. Then lim f(x) =  x -> a means that the values of f(x) can be made arbitrarily large by taking x close enough to a.

4. Another notation for lim x - > a f(x) =  is “f(x) -- >  as x -- > a” For such a limit, we say: “the limit of f(x), as x approaches a, is infinity” “f(x) approaches infinity as x approaches a” “f(x) increases without bound as x approaches a”

5. What about f(x) = 1/x, as x -- > 0 ?

6. Definition The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true: lim f(x) =  lim f(x) = -  lim f(x) = - . x -- > a + x -- > a - x -- > a +

7. Example:

8. Example Find the vertical asymptotes of f(x) = ln (x – 5).

9. CHAPTER 2 2.4 Continuity Sec 2.6: Limits at Infinity f(x) = (x 2 -1) / (x 2 +1) f(x) = e x

10. CHAPTER 2 2.4 Continuity 4 Sec 2.6: Limits at Infinity f(x) = tan -1 x f(x) = 1/x

11. CHAPTER 2 2.4 Continuity Sec 2.6: Limits at Infinity animation http://math.sfsu.edu/goetz/Teaching/math226f00/animations/limit.mov

12. Let f be a function defined on some interval (a,  ). Then lim f (x) = L x - >  means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Definition: Limit at Infinity

13. Definition The line y = L is called a horizontal asymptote of the curve y = f(x) if either lim f(x) = L or lim f(x) = L. x ->  x -> -  lim tan - 1(x)= -  /2 x -> -  lim tan –1 (x) =  /2. x -> 

14. If n is a positive integer, then lim 1/ x n = 0 lim 1/ x n = 0. x-> -  x-> -  lim e x = 0. x-> - 

15. Example lim (7t 3 + 4t ) / (2t 3 - t 2 + 3). x-> - 

16. We know lim x-> -  e x = 0. What about lim x->  e x ? f(x) = e x

17. So lim t ->  Ae rt =  for any r > 0. Say P(t) = Ae rt represents a population at time t. This is a mathematical model of “exponential growth,” where r is the growth rate and A is the initial population. See http://cauchy.math.colostate.edu/Applets Exponential Growth Model

18. Exponential growth (r > 0) Exponential decay (r < 0) For f(t) = Ae rt : Exponential Growth/Decay

19. A more complicated model of population growth is the logistic equation: P(t) = K / (1 + Ae –rt ) What is lim t ->  P(t) ? In this model, K represents a “carrying capacity”: the maximum population that the environment is capable of sustaining. Logistic Growth Model

20. Logistic equation as a model of yeast growth http://www-rohan.sdsu.edu/~jmahaffy/ Logistic Growth Model