Today: Limits Involving Infinity lim f(x) = x -> a Infinite limits Limits at infinity lim f(x) = L...
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Today: Limits Involving Infinity -4 -3 -2 -1 1 2 3 4 10 20 30 40 f H x L = 1 €€€€€€ x 2 lim f(x) = x -> a Infinite limits Limits at infinity lim f(x) = L x ->
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Transcript of Today: Limits Involving Infinity lim f(x) = x -> a Infinite limits Limits at infinity lim f(x) = L...
Today: Limits Involving Infinity
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fHxL= 1€€€€€€x2
lim f(x) =
x -> a
Infinite limits Limits at infinity
lim f(x) = L
x ->
CHAPTER 2 2.4 Continuity
Infinite Limits
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fHxL= 1€€€€€€€€€€€€€€€€€€€Hx - 2L2
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fHxL= 1€€€€€€x2
(see Sec 2.2, pp 98-101)
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.
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fHxL= 1€€€€€€€€€€€€€€€€€€€Hx - 2L2
Another notation for lim x -> a f(x) = is
“f(x) --> as x --> a”
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fHxL= 1€€€€€€x2
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”
What about f(x) = 1/x, as x --> 0 ?
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 +
Example:
Example Find the vertical asymptotes of f(x) = ln (x – 5).
CHAPTER 2 2.4 Continuity
Sec 2.6: Limits at Infinity
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fHxL= x2 - 1€€€€€€€€€€€€€€x2 +1
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fHxL=exf(x) = (x2-1) / (x2 +1)
f(x) = ex
CHAPTER 2 2.4 Continuity
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Sec 2.6: Limits at Infinity
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fHxL=tan- 1HxL
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fHxL= 1€€€€x
f(x) = tan-1 x
f(x) = 1/x
CHAPTER 2 2.4 Continuity
Sec 2.6: Limits at Infinity
animation
http://math.sfsu.edu/goetz/Teaching/math226f00/animations/limit.mov
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
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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fHxL=tan- 1HxL
If n is a positive integer, then
lim 1/ x n = 0 lim 1/ x n = 0. x-> - x-> -
lim e x = 0. x-> -
Example lim (7t 3 + 4t ) / (2t 3 - t 2+ 3).
x-> -
• We know lim x-> - e x = 0.
• What about lim x-> e x ?
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fHxL=ex
f(x) = ex
• 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
• Exponential growth (r > 0)
• Exponential decay (r < 0)
For f(t) = Ae rt :
Exponential Growth/Decay
• 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
• Logistic equation as a model of yeast growth
http://www-rohan.sdsu.edu/~jmahaffy/
Logistic Growth Model
