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

-4 -3 -2 -1 1 2 3 4

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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.

-4 -3 -2 -1 1 2 3 4

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fHxL= 1€€€€€€€€€€€€€€€€€€€Hx - 2L2

Another notation for lim x -> a f(x) = is

“f(x) --> as x --> a”

-4 -3 -2 -1 1 2 3 4

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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

4

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