Math 3
02/10
Section 2
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Infinite Limits
What is
lim
x→∞
x2 + 1
=
x2 − 1
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Limits at infinity
Let f be a function that is defined on some interval (a, ∞)
(resp. (−∞, a)). Then
lim f (x) = L
x→∞
(resp. lim f (x) = L)
x→−∞
if the values of f (x) can be made arbitrarily close to L by requiring x to be
sufficiently large. (resp. sufficiently large negative)
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Theorem
If r > 0 is a rational number, then
lim
x→∞
1
=0
xr
If r > 0 is a rational number such that x r is defined for all x, then
1
=0
x→−∞ x r
lim
Ex:
lim
x→−∞
1
= 0
x2
1
lim √ = 0
x
x→∞
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3x 2 − x − 2
Example: lim
x→∞ 5x 2 + 4x + 1
3x 2 − x − 2
lim
x→∞ 5x 2 + 4x + 1
=
lim
x→∞
3 −
5 +
1
x
4
x
−
+
=
3x 2 − x − 2
lim
·
x→∞ 5x 2 + 4x + 1
1
x2
1
x2
2
x2
1
x2
=
3 − 0 − 0
5 + 0 + 0
3
5
=
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Examples
√
1
Find lim
x→−∞
2
Find lim
x→∞
x + 3x 2
4x − 1
p
x2 + 1 − x
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Solutions
√
1
Find lim
x→−∞
2
Find lim
x→∞
√
x + 3x 2
− 3
=
4x − 1
4
p
x2 + 1 − x = 0
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Horizontal Asymptote
The line y = L is called a horizontal asymptote of the curve y = f (x) if
either
lim f (x) = L
x→∞
lim f (x) = L
x→−∞
Ex: y = arctan(x) has a horizontal asymptotes at
π
2
and
−π
2
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Infinite limits at infinity
The notation
lim f (x) = ∞
x→∞
(resp. lim f (x) = −∞)
x→∞
Is used to indicate that the function gets arbitrarily large (resp. large
negative) as x gets large.
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Examples: Infinite limits at infinity
Figure: lim −x 3 = −∞
x→∞
Figure: lim e x = ∞
x→∞
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Example
Find the horizontal and vertical asymptotes of
(
f (x) =
x 3 +x
1−x
4 + sin
1
x
x <1
x ≥1
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(
Graph of f (x) =
x 3 +x
1−x
4 + sin
1
x
x <1
x ≥1
Vertical asymptote x = 1
Horizontal asymptote y = 4
(Red)
(Purple)
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Practice
Find the horizontal and vertical asymptotes of
√
f (x) =
2x 2 + 1
3x − 5
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Formal definition of a limit at infinity
Let f be a function defined on some interval (a, ∞). Then
lim f (x) = L
x→∞
means that for every > 0 there is a corresponding positive number N
such that
if
x >N
then
|f (x) − L| < February 10, 2017
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Formal definition of a limit at infinity
Let f be a function defined on some interval (a, ∞). Then
lim f (x) = L
x→∞
means that for every positive number M there is a corresponding positive
number N such that
if
x >N
then
f (x) > M
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