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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.

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Today: Limits Involving Infinity

Infinite limits

Limits at infinity

lim f(x) = L

x -> 

lim f(x) = 

x -> a


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Infinite Limits

CHAPTER 2

2.4 Continuity

(see Sec 2.2, pp 98-101)


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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


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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”


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


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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 -



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


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Sec 2.6: Limits at Infinity

CHAPTER 2

2.4 Continuity

f(x) = (x2-1) / (x2 +1)

f(x) = ex


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4

Sec 2.6: Limits at Infinity

CHAPTER 2

2.4 Continuity

f(x) = tan-1 x

f(x) = 1/x


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Sec 2.6: Limits at Infinity

CHAPTER 2

2.4 Continuity

http://math.sfsu.edu/goetz/Teaching/math226f00/animations/limit.mov

animation


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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.


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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 -> 


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If n is a positive integer, then

lim 1/ x n = 0 lim 1/ x n = 0. x-> -  x-> - 

lim e x = 0. x-> - 


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Example lim (7t 3 + 4t ) / (2t 3 - t 2+ 3).

x-> - 


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f(x) = ex


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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


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Exponential Growth/Decay

Forf(t) = Ae rt :

  • Exponential growth (r > 0)

  • Exponential decay (r < 0)


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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.


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Logistic Growth Model

  • Logistic equation as a model of yeast growth

http://www-rohan.sdsu.edu/~jmahaffy/