Lesson 4 Limits Involving Infinity 1202826891350145 2

8/9/2019 Lesson 4 Limits Involving Infinity 1202826891350145 2

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Section 2.5Limits Involving Innity

Math 1a

February 4, 2008

Announcements

Syllabus available on course websiteAll HW on website nowNo class Monday 2/18ALEKS due Wednesday 2/20
http://my.harvard.edu/k30281http://my.harvard.edu/k30281http://find/http://goback/


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Outline

Innite LimitsVertical AsymptotesInnite Limits we KnowLimit Laws with Innite LimitsIndeterminate Limits

Limits at InnityAlgebraic rates of growthExponential rates of growth

Rationalizing to get a limit

Worksheet
http://find/http://goback/


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Innite Limits

DenitionThe notation

limx a

f (x ) = means that the values of f (x ) can be made arbitrarily large (aslarge as we please) by taking x sufficiently close toa but not equalto a.

DenitionThe notation

limx a

f (x ) =

means that the values of f (x ) can be made arbitrarily largenegative by taking x sufficiently close toa but not equal to a.Of course we have denitions for left- and right-hand innite limits.


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Vertical Asymptotes

DenitionThe line x = a is called a vertical asymptote of the curvey = f (x ) if at least one of the following is true:

limx a

f (x ) = lim

x a +f (x ) =

limx a

f (x ) =

limx a

f (x ) = lim

x a +f (x ) =

limx a

f (x ) =


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Finding limits at trouble spots

ExampleLet

f (t ) =t 2 + 2

t 2

3t + 2

Find limt a

f (t ) and limt a +

f (t ) for each a at which f is notcontinuous.


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Finding limits at trouble spots

ExampleLet

f (t ) =t 2 + 2

t 2

3t + 2

Find limt a

f (t ) and limt a +

f (t ) for each a at which f is notcontinuous.

Solution

The denominator factors as (t 1)( t 2). We can record the signs of the factors on the number line.


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(t 1) 10 +


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(t 1) 10 +

(t 2) 20 +


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(t 1) 10 +

(t 2) 20 +

(t 2 + 2)+

f (t )1 2


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(t 1) 10 +

(t 2) 20 +

(t 2 + 2)+

f (t )1 2

+


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(t 1) 10 +

(t 2) 20 +

(t 2 + 2)+

f (t )1 2

+


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(t 1) 10 +

(t 2) 20 +

(t 2 + 2)+

f (t )1 2

+


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(t 1) 10 +

(t 2) 20 +

(t 2 + 2)+

f (t )1 2

+


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(t 1) 10 +

(t 2) 20 +

(t 2 + 2)+

f (t )1 2

+ +


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Limit Laws with innite limitsTo aid your intuition

The sum of positive innite limits is. That is

+ = The sum of negative innite limits is.

= The sum of a nite limit and an innite limit is innite.

a + = a =


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Rules of Thumb with innite limitsDont try this at home!

The sum of positive innite limits is. That is

+ = The sum of negative innite limits is

.

= The sum of a nite limit and an innite limit is innite.

a + = a =


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Rules of Thumb with innite limitsThe product of a nite limit and an innite limit is inniteif the nite limit is not 0.

a = if a > 0 if a < 0.a () =

if a > 0

if a < 0.

The product of two innite limits is innite.

= () =

() () = The quotient of a nite limit by an innite limit is zero:

a

= 0 .


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Indeterminate Limits

Limits of the form 0and are indeterminate . Thereis no rule for evaluating such a form; the limit must beexamined more closely.

d


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Indeterminate Limits

Limits of the form 0and are indeterminate . Thereis no rule for evaluating such a form; the limit must beexamined more closely.Limits of the form

10

are also indeterminate.

O li


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Outline


Limits at InnityAlgebraic rates of growthExponential rates of growth


Worksheet


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DenitionLet f be a function dened on some interval (a, ). Then

limx

f (x ) = L

means that the values of f (x ) can be made as close to L as welike, by taking x sufficiently large.


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limx

f (x ) = L


DenitionThe line y = L is a called a horizontal asymptote of the curvey = f (x ) if either

limx f (x ) = L or limx f (x ) = L.


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limx

f (x ) = L


DenitionThe line y = L is a called a horizontal asymptote of the curvey = f (x ) if either

limx f (x ) = L or limx f (x ) = L.

y = L is a horizontal line!


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TheoremLet n be a positive integer. Then

limx

1

x n = 0

limx

1x n

= 0

Using the limit laws to compute limits at


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ExampleFind

limx

2x 3 + 3 x + 14x 3 + 5 x 2 + 7

if it exists.A does not existB 1/ 2

C 0D



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ExampleFind

limx

2x 3 + 3 x + 14x 3 + 5 x 2 + 7

if it exists.A does not existB 1/ 2

C 0D


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SolutionFactor out the largest power of x from the numerator and

denominator. We have 2x 3 + 3 x + 14x 3 + 5 x 2 + 7

=x 3 (2 + 3/ x 2 + 1/ x 3 )x 3 (4 + 5/ x + 7/ x 3 )

limx

2x 3 + 3 x + 1

4x 3 + 5 x 2 + 7= lim

x

2 + 3/ x 2 + 1/ x 3

4 + 5/ x + 7/ x 3

=2 + 0 + 04 + 0 + 0

=12


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SolutionFactor out the largest power of x from the numerator and

denominator. We have 2x 3 + 3 x + 14x 3 + 5 x 2 + 7

=x 3 (2 + 3/ x 2 + 1/ x 3 )x 3 (4 + 5/ x + 7/ x 3 )

limx

2x 3 + 3 x + 1

4x 3 + 5 x 2 + 7= lim

x

2 + 3/ x 2 + 1/ x 3

4 + 5/ x + 7/ x 3

=2 + 0 + 04 + 0 + 0

=12

UpshotWhen nding limits of algebraic expressions at innitely, look atthe highest degree terms.

Another Example


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Another Example

ExampleFind

limx 3x 4 + 7

x 2 + 3

Another Example


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Another Example

ExampleFind

limx 3x 4 + 7

x 2 + 3

SolutionThe limit is 3.


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Example

Make a conjecture about limx

x 2

2x .


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Example

Make a conjecture about limx

x 2

2x .

SolutionThe limit is zero. Exponential growth is innitely faster thangeometric growth



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g g

ExampleCompute lim

x

4x 2 + 17 2x .



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g g

ExampleCompute lim

x

4x 2 + 17 2x .

SolutionThis limit is of the form , which we cannot use. So we rationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.

Outline


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Limits at InnityAlgebraic rates of growthExponential rates of growthRationalizing to get a limit

Worksheet


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Worksheet

Lesson 4 Limits Involving Infinity 1202826891350145 2

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