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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
Infinite Limits CHAPTER 2 2.4 Continuity (see Sec 2.2, pp 98-101)
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
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”
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 -
Sec 2.6: Limits at Infinity CHAPTER 2 2.4 Continuity f(x) = (x2-1) / (x2 +1) f(x) = ex
4 Sec 2.6: Limits at Infinity CHAPTER 2 2.4 Continuity f(x) = tan-1 x f(x) = 1/x
Sec 2.6: Limits at Infinity CHAPTER 2 2.4 Continuity http://math.sfsu.edu/goetz/Teaching/math226f00/animations/limit.mov animation
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.
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 ->
If n is a positive integer, then lim 1/ x n = 0 lim 1/ x n = 0. x-> - x-> - lim e x = 0. x-> -
We know lim x-> - e x = 0. • What about lim x-> e x ? f(x) = ex
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
Exponential Growth/Decay Forf(t) = Ae rt : • Exponential growth (r > 0) • Exponential decay (r < 0)
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.
Logistic Growth Model • Logistic equation as a model of yeast growth http://www-rohan.sdsu.edu/~jmahaffy/