Section 4.4 Limits at Infinity; Horizontal Asymptotes AP Calculus November 2, 2009 Berkley High...
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Section 4.4 Limits at Infinity; Horizontal Asymptotes AP Calculus November 2, 2009 Berkley High School, D1B1 [email protected]
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Transcript of Section 4.4 Limits at Infinity; Horizontal Asymptotes AP Calculus November 2, 2009 Berkley High...
Section 4.4Limits at Infinity; Horizontal AsymptotesAP CalculusNovember 2, 2009Berkley High School, [email protected]
Calculus, Section 4.4, Todd Fadoir 2
Limit at Infinity
How do we find the value of a function at a point that doesn’t exist?
We truly can’t, so we use the idea of approaching infinity as compared to being at infinity.
Calculus, Section 4.4, Todd Fadoir 3
Limit at Infinity
The limit of f(x) as x approaches infinity equals L means that the values of f(x) can be made arbitrarily close to L by making x sufficiently large.
How close? Calculus close.
lim ( )x
f x L
Calculus, Section 4.4, Todd Fadoir 4
Limit at Negative Infinity
The limit of f(x) as x approaches negative infinity equals L means that the values of f(x) can be made arbitrarily close to L by making x sufficiently large and negative.
lim ( )x
f x L
Calculus, Section 4.4, Todd Fadoir 5
Horizontal Asymptote
The line is called a horizontal asymptote
of ( ) if either
lim ( )
or
lim ( )
x
x
y L
y f x
f x L
f x L
Calculus, Section 4.4, Todd Fadoir 6
Limits that go to zero
1lim 0, if 0 and is rational.
1lim 0, if 0 and is rational.
rx
rx
r rx
r rx
Calculus, Section 4.4, Todd Fadoir 7
Example
2
2
1lim
1x
x
x
Attack 1: Common Sense
As x get really big, the “-1” and the “+1” are so small they we can forget about them.
We are left with x2/x2. This equals 1.
The problem with this attack is that, while it make sense, what rules of algebra did we follow? We put this “proof” in a two column form, what reasons would we give?
Calculus, Section 4.4, Todd Fadoir 8
Example
2
2
22
22
1lim
11
1lim
11
x
x
x
x
xx
xx
Attack 2: Algebraic
2
2 2
2
2 2
2
2
1
lim1
11 1
lim 11 11
x
x
xx xxx x
x
x
Conclusion: there is a horizontal asymptote at y =1
Calculus, Section 4.4, Todd Fadoir 9
Example
2lim( ) lim( ( 1))x x
x x x x
Calculus, Section 4.4, Todd Fadoir 10
Example2
2
2
2 2
2
12 12 1
lim lim13 5 (3 5)
2 1
lim3 5
12
lim5
3
2
3
x x
x
x
xx xx x
x
xx xxx x
x
x
Calculus, Section 4.4, Todd Fadoir 11
Assignment
Section 4.4, 1-35, 39, odd
