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Today: Limits Involving Infinity

Infinite limits

Limits at infinity

lim f(x) = L

x ->

lim f(x) =

x -> a

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 -

CHAPTER 2

2.4 Continuity

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

animation

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 ->

- 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

- 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 equation as a model of yeast growth

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

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