Section 2.5 Limits at Infinity Math 1a October 10, 2007 Announcements I Midterm I is coming: October 24, 7:00-9:00 in Halls A and C
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Section 2.5Limits at Infinity
Math 1a
October 10, 2007
Announcements
I Midterm I is coming: October 24, 7:00-9:00 in Halls A and C
Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007
Page1of8
DefinitionLet f be a function defined 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.
DefinitionThe 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!
DefinitionLet f be a function defined 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.
DefinitionThe 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!
DefinitionLet f be a function defined 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.
DefinitionThe 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!
TheoremLet n be a positive integer. Then
I limx→∞
1
xn= 0
I limx→−∞
1
xn= 0
Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007
Page3of8
Using the limit laws to compute limits at ∞
Example
Find
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
Using the limit laws to compute limits at ∞
Example
Find
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
SolutionFactor out the largest power of x from the numerator anddenominator. We have
2x3 + 3x + 1
4x3 + 5x2 + 7=
x3(2 + 3/x2 + 1/x3)
x3(4 + 5/x + 7/x3)
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7= lim
x→∞
2 + 3/x2 + 1/x3
4 + 5/x + 7/x3
=2 + 0 + 0
4 + 0 + 0=
1
2
Upshot
When finding limits of algebraic expressions at infinitely, look atthe highest degree terms.
SolutionFactor out the largest power of x from the numerator anddenominator. We have
2x3 + 3x + 1
4x3 + 5x2 + 7=
x3(2 + 3/x2 + 1/x3)
x3(4 + 5/x + 7/x3)
limx→∞
2x3 + 3x + 1
4x3 + 5x2 + 7= lim
x→∞
2 + 3/x2 + 1/x3
4 + 5/x + 7/x3
=2 + 0 + 0
4 + 0 + 0=
1
2
Upshot
When finding limits of algebraic expressions at infinitely, look atthe highest degree terms.
Another Example
Example
Find
limx→∞
√3x4 + 7
x2 + 3
SolutionThe limit is
√3.
Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007
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Another Example
Example
Find
limx→∞
√3x4 + 7
x2 + 3
SolutionThe limit is
√3.
Example
Make a conjecture about limx→∞
x2
2x.
SolutionThe limit is zero. Exponential growth is infinitely faster thangeometric growth
Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007
Page6of8
Example
Make a conjecture about limx→∞
x2
2x.
SolutionThe limit is zero. Exponential growth is infinitely faster thangeometric growth
Rationalizing to get a limit
Example
Compute limx→∞
(√4x2 + 17− 2x
).
SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.
Math 1a - October 10, 2007.GWBWednesday, Oct 10, 2007
Page7of8
Rationalizing to get a limit
Example
Compute limx→∞
(√4x2 + 17− 2x
).
SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.
Worksheet
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