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
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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
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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
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
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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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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.
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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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.
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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Using the limit laws to compute limits at
ExampleFind
limx
2x 3 + 3 x + 14x 3 + 5 x 2 + 7
if it exists.A does not existB 1/ 2
C 0D
Using the limit laws to compute limits at
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Using the limit laws to compute limits at
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
Rationalizing to get a limit
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g g
ExampleCompute lim
x
4x 2 + 17 2x .
Rationalizing to get a limit
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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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Innite LimitsVertical AsymptotesInnite Limits we KnowLimit Laws with Innite LimitsIndeterminate Limits
Limits at InnityAlgebraic rates of growthExponential rates of growthRationalizing to get a limit
Worksheet
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Worksheet
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