1 / 20

Today: Limits Involving Infinity

Today: Limits Involving Infinity . Infinite limits. Limits at infinity. lim f(x) = L x -> . lim f(x) =  x -> a. Infinite Limits. CHAPTER 2. 2.4 Continuity. (see Sec 2.2, pp 98-101) . CHAPTER 2.

Antony
Download Presentation

Today: Limits Involving Infinity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

  3. CHAPTER 2 DefinitionLet f be afunction definedon 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. 2.4 Continuity

  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) = -  lim f(x) = - . x --> a - x --> a+ x --> a + x --> a -

  7. Example:

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

  9. Sec 2.6: Limits at Infinity CHAPTER 2 2.4 Continuity f(x) = (x2-1) / (x2 +1) f(x) = ex

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

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

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

  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) = ex

  17. Exponential Growth Model • 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

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

  19. Logistic Growth Model • 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.

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

More Related