lesson 6: limits involving ∞ (section 41 slides)
TRANSCRIPT
Section 1.6Limits involving Infinity
V63.0121.041, Calculus I
New York University
September 22, 2010
Announcements
I Quiz 1 is next week in recitation. Covers Sections 1.1–1.4
. . . . . .
. . . . . .
Announcements
I Quiz 1 is next week inrecitation. Covers Sections1.1–1.4
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 2 / 37
. . . . . .
Objectives
I “Intuit” limits involvinginfinity by eyeballing theexpression.
I Show limits involvinginfinity by algebraicmanipulation andconceptual argument.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 3 / 37
. . . . . .
Recall the definition of limit
DefinitionWe write
limx→a
f(x) = L
and say
“the limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L aswe like) by taking x to be sufficiently close to a (on either side of a) butnot equal to a.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 4 / 37
. . . . . .
Recall the unboundedness problem
Recall why limx→0+
1xdoesn’t exist.
. .x
.y
..L?
No matter how thin we draw the strip to the right of x = 0, we cannot“capture” the graph inside the box.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 5 / 37
. . . . . .
Recall the unboundedness problem
Recall why limx→0+
1xdoesn’t exist.
. .x
.y
..L?
No matter how thin we draw the strip to the right of x = 0, we cannot“capture” the graph inside the box.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 5 / 37
. . . . . .
Recall the unboundedness problem
Recall why limx→0+
1xdoesn’t exist.
. .x
.y
..L?
No matter how thin we draw the strip to the right of x = 0, we cannot“capture” the graph inside the box.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 5 / 37
. . . . . .
Recall the unboundedness problem
Recall why limx→0+
1xdoesn’t exist.
. .x
.y
..L?
No matter how thin we draw the strip to the right of x = 0, we cannot“capture” the graph inside the box.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 5 / 37
. . . . . .
Outline
Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limit forms
Limits at ∞Algebraic rates of growthRationalizing to get a limit
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 6 / 37
. . . . . .
Infinite Limits
DefinitionThe notation
limx→a
f(x) = ∞
means that values of f(x) canbe made arbitrarily large (aslarge as we please) by taking xsufficiently close to a but notequal to a.
I “Large” takes the place of“close to L”.
. .x
.y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 37
. . . . . .
Infinite Limits
DefinitionThe notation
limx→a
f(x) = ∞
means that values of f(x) canbe made arbitrarily large (aslarge as we please) by taking xsufficiently close to a but notequal to a.
I “Large” takes the place of“close to L”.
. .x
.y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 37
. . . . . .
Infinite Limits
DefinitionThe notation
limx→a
f(x) = ∞
means that values of f(x) canbe made arbitrarily large (aslarge as we please) by taking xsufficiently close to a but notequal to a.
I “Large” takes the place of“close to L”.
. .x
.y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 37
. . . . . .
Infinite Limits
DefinitionThe notation
limx→a
f(x) = ∞
means that values of f(x) canbe made arbitrarily large (aslarge as we please) by taking xsufficiently close to a but notequal to a.
I “Large” takes the place of“close to L”.
. .x
.y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 37
. . . . . .
Infinite Limits
DefinitionThe notation
limx→a
f(x) = ∞
means that values of f(x) canbe made arbitrarily large (aslarge as we please) by taking xsufficiently close to a but notequal to a.
I “Large” takes the place of“close to L”.
. .x
.y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 37
. . . . . .
Infinite Limits
DefinitionThe notation
limx→a
f(x) = ∞
means that values of f(x) canbe made arbitrarily large (aslarge as we please) by taking xsufficiently close to a but notequal to a.
I “Large” takes the place of“close to L”.
. .x
.y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 37
. . . . . .
Infinite Limits
DefinitionThe notation
limx→a
f(x) = ∞
means that values of f(x) canbe made arbitrarily large (aslarge as we please) by taking xsufficiently close to a but notequal to a.
I “Large” takes the place of“close to L”.
. .x
.y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 37
. . . . . .
Infinite Limits
DefinitionThe notation
limx→a
f(x) = ∞
means that values of f(x) canbe made arbitrarily large (aslarge as we please) by taking xsufficiently close to a but notequal to a.
I “Large” takes the place of“close to L”.
. .x
.y
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 7 / 37
. . . . . .
Negative Infinity
DefinitionThe notation
limx→a
f(x) = −∞
means that the values of f(x) can be made arbitrarily large negative (aslarge as we please) by taking x sufficiently close to a but not equal to a.
I We call a number large or small based on its absolute value. So−1,000,000 is a large (negative) number.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 8 / 37
. . . . . .
Negative Infinity
DefinitionThe notation
limx→a
f(x) = −∞
means that the values of f(x) can be made arbitrarily large negative (aslarge as we please) by taking x sufficiently close to a but not equal to a.
I We call a number large or small based on its absolute value. So−1,000,000 is a large (negative) number.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 8 / 37
. . . . . .
Vertical Asymptotes
DefinitionThe line x = a is called a vertical asymptote of the curve y = f(x) if atleast one of the following is true:
I limx→a
f(x) = ∞
I limx→a+
f(x) = ∞
I limx→a−
f(x) = ∞
I limx→a
f(x) = −∞
I limx→a+
f(x) = −∞
I limx→a−
f(x) = −∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 9 / 37
. . . . . .
Infinite Limits we Know
I limx→0+
1x= ∞
I limx→0−
1x= −∞
I limx→0
1x2
= ∞
. .x
.y
.
.
.
.
.
.
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.
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 10 / 37
. . . . . .
Infinite Limits we Know
I limx→0+
1x= ∞
I limx→0−
1x= −∞
I limx→0
1x2
= ∞
. .x
.y
.
.
.
.
.
.
.
.
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 10 / 37
. . . . . .
Infinite Limits we Know
I limx→0+
1x= ∞
I limx→0−
1x= −∞
I limx→0
1x2
= ∞
. .x
.y
.
.
.
.
.
.
.
.
.
.
.
.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 10 / 37
. . . . . .
Finding limits at trouble spots
Example
Let
f(x) =x2 + 2
x2 − 3x+ 2Find lim
x→a−f(x) and lim
x→a+f(x) for each a at which f is not continuous.
SolutionThe denominator factors as (x− 1)(x− 2). We can record the signs ofthe factors on the number line.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 11 / 37
. . . . . .
Finding limits at trouble spots
Example
Let
f(x) =x2 + 2
x2 − 3x+ 2Find lim
x→a−f(x) and lim
x→a+f(x) for each a at which f is not continuous.
SolutionThe denominator factors as (x− 1)(x− 2). We can record the signs ofthe factors on the number line.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 11 / 37
. . . . . .
Use the number line
. .(x− 1)
.− ..1.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+
.+∞ .−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞
.−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞
.− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .−
.−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞
.+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞
.+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
So
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
Solim
x→1−f(x) = +∞
limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
Solim
x→1−f(x) = +∞
limx→2−
f(x) = −∞
limx→1+
f(x) = −∞
limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
Solim
x→1−f(x) = +∞ lim
x→2−f(x) = −∞
limx→1+
f(x) = −∞
limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
Use the number line
. .(x− 1).− .
.1
.0 .+
.(x− 2).− .
.2
.0 .+
.(x2 + 2).+
.f(x)..1
..2
.+ .+∞ .−∞ .− .−∞ .+∞ .+
Solim
x→1−f(x) = +∞ lim
x→2−f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 12 / 37
. . . . . .
In English, now
To explain the limit, you can say:“As x → 1−, the numerator approaches 3, and the denominatorapproaches 0 while remaining positive. So the limit is +∞.”
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 13 / 37
. . . . . .
The graph so far
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
. .x
.y
..−1
..1
..2
..3
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 14 / 37
. . . . . .
The graph so far
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
. .x
.y
..−1
..1
..2
..3
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 14 / 37
. . . . . .
The graph so far
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
. .x
.y
..−1
..1
..2
..3
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 14 / 37
. . . . . .
The graph so far
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
. .x
.y
..−1
..1
..2
..3
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 14 / 37
. . . . . .
The graph so far
limx→1−
f(x) = +∞ limx→2−
f(x) = −∞
limx→1+
f(x) = −∞ limx→2+
f(x) = +∞
. .x
.y
..−1
..1
..2
..3
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 14 / 37
. . . . . .
Limit Laws (?) with infinite limits
Fact
I If limx→a
f(x) = ∞ and limx→a
g(x) = ∞,
then limx→a
(f(x) + g(x)) = ∞.
.
.∞+∞ = ∞
I If limx→a
f(x) = −∞ andlimx→a
g(x) = −∞, then
limx→a
(f(x) + g(x)) = −∞.
.
.−∞+ (−∞) = −∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 15 / 37
. . . . . .
Rules of Thumb with infinite limits
Fact
I If limx→a
f(x) = ∞ and limx→a
g(x) = ∞,
then limx→a
(f(x) + g(x)) = ∞..
.∞+∞ = ∞
I If limx→a
f(x) = −∞ andlimx→a
g(x) = −∞, then
limx→a
(f(x) + g(x)) = −∞.
.
.−∞+ (−∞) = −∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 15 / 37
. . . . . .
Rules of Thumb with infinite limits
Fact
I If limx→a
f(x) = ∞ and limx→a
g(x) = ∞,
then limx→a
(f(x) + g(x)) = ∞..
.∞+∞ = ∞
I If limx→a
f(x) = −∞ andlimx→a
g(x) = −∞, then
limx→a
(f(x) + g(x)) = −∞..
.−∞+ (−∞) = −∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 15 / 37
. . . . . .
Rules of Thumb with infinite limits
Fact
I If limx→a
f(x) = L and limx→a
g(x) = ±∞,
.
.L+∞ = ∞L−∞ = −∞
then limx→a
(f(x) + g(x)) = ±∞.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 16 / 37
. . . . . .
Rules of Thumb with infinite limits
Fact
I If limx→a
f(x) = L and limx→a
g(x) = ±∞,.
.L+∞ = ∞L−∞ = −∞
then limx→a
(f(x) + g(x)) = ±∞.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 16 / 37
. . . . . .
Rules of Thumb with infinite limitsKids, don't try this at home!
Fact
I The product of a finite limit and an infinite limit is infinite if the finitelimit is not 0.
.
.L · ∞ =
{∞ if L > 0−∞ if L < 0.
.
.L · (−∞) =
{−∞ if L > 0∞ if L < 0.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 17 / 37
. . . . . .
Rules of Thumb with infinite limitsKids, don't try this at home!
Fact
I The product of a finite limit and an infinite limit is infinite if the finitelimit is not 0. .
.L · ∞ =
{∞ if L > 0−∞ if L < 0.
.
.L · (−∞) =
{−∞ if L > 0∞ if L < 0.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 17 / 37
. . . . . .
Rules of Thumb with infinite limitsKids, don't try this at home!
Fact
I The product of a finite limit and an infinite limit is infinite if the finitelimit is not 0. .
.L · ∞ =
{∞ if L > 0−∞ if L < 0.
.
.L · (−∞) =
{−∞ if L > 0∞ if L < 0.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 17 / 37
. . . . . .
Multiplying infinite limitsKids, don't try this at home!
Fact
I The product of two infinite limits is infinite.
.
.∞ ·∞ = ∞
∞ · (−∞) = −∞(−∞) · (−∞) = ∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 18 / 37
. . . . . .
Multiplying infinite limitsKids, don't try this at home!
Fact
I The product of two infinite limits is infinite. .
.∞ ·∞ = ∞
∞ · (−∞) = −∞(−∞) · (−∞) = ∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 18 / 37
. . . . . .
Dividing by InfinityKids, don't try this at home!
Fact
I The quotient of a finite limit by an infinite limit is zero.
.
.L∞
= 0
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 19 / 37
. . . . . .
Dividing by InfinityKids, don't try this at home!
Fact
I The quotient of a finite limit by an infinite limit is zero..
.L∞
= 0
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 19 / 37
. . . . . .
Dividing by zero is still not allowed
..10= ∞
There are examples of such limit forms where the limit is ∞, −∞,undecided between the two, or truly neither.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 20 / 37
. . . . . .
Indeterminate Limit forms
Limits of the formL0are indeterminate. There is no rule for evaluating
such a form; the limit must be examined more closely. Consider these:
limx→0
1x2
= ∞ limx→0
−1x2
= −∞
limx→0+
1x= ∞ lim
x→0−1x= −∞
Worst, limx→0
1x sin(1/x)
is of the formL0, but the limit does not exist, even
in the left- or right-hand sense. There are infinitely many verticalasymptotes arbitrarily close to 0!
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 21 / 37
. . . . . .
Indeterminate Limit forms
Limits of the form 0 · ∞ and ∞−∞ are also indeterminate.
Example
I The limit limx→0+
sin x · 1xis of the form 0 · ∞, but the answer is 1.
I The limit limx→0+
sin2 x · 1xis of the form 0 · ∞, but the answer is 0.
I The limit limx→0+
sin x · 1x2
is of the form 0 · ∞, but the answer is ∞.
Limits of indeterminate forms may or may not “exist.” It will depend onthe context.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 22 / 37
. . . . . .
Indeterminate forms are like Tug Of War
Which side wins depends on which side is stronger.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 23 / 37
. . . . . .
Outline
Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limit forms
Limits at ∞Algebraic rates of growthRationalizing to get a limit
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 24 / 37
. . . . . .
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 we like, bytaking x sufficiently large.
DefinitionThe line y = L is a called a horizontal asymptote of the curve y = f(x)if either
limx→∞
f(x) = L or limx→−∞
f(x) = L.
y = L is a horizontal line!
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 25 / 37
. . . . . .
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 we like, bytaking x sufficiently large.
DefinitionThe line y = L is a called a horizontal asymptote of the curve y = f(x)if either
limx→∞
f(x) = L or limx→−∞
f(x) = L.
y = L is a horizontal line!
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 25 / 37
. . . . . .
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 we like, bytaking x sufficiently large.
DefinitionThe line y = L is a called a horizontal asymptote of the curve y = f(x)if either
limx→∞
f(x) = L or limx→−∞
f(x) = L.
y = L is a horizontal line!
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 25 / 37
. . . . . .
Basic limits at infinity
TheoremLet n be a positive integer. Then
I limx→∞
1xn
= 0
I limx→−∞
1xn
= 0
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 26 / 37
. . . . . .
Limit laws at infinity
FactAny limit law that concerns finite limits at a finite point a is still true ifthe finite point is replaced by infinity.That is, if lim
x→∞f(x) = L and lim
x→∞g(x) = M, then
I limx→∞
(f(x) + g(x)) = L+M
I limx→∞
(f(x)− g(x)) = L−M
I limx→∞
cf(x) = c · L (for any constant c)
I limx→∞
f(x) · g(x) = L ·M
I limx→∞
f(x)g(x)
=LM
(if M ̸= 0)
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 27 / 37
. . . . . .
Using the limit laws to compute limits at ∞
Example
Find limx→∞
xx2 + 1
AnswerThe limit is 0.
. .x
.y
Notice that the graph does cross the asymptote, which contradicts oneof the commonly held beliefs of what an asymptote is.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 28 / 37
. . . . . .
Using the limit laws to compute limits at ∞
Example
Find limx→∞
xx2 + 1
AnswerThe limit is 0.
. .x
.y
Notice that the graph does cross the asymptote, which contradicts oneof the commonly held beliefs of what an asymptote is.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 28 / 37
. . . . . .
Solution
SolutionFactor out the largest power of x from the numerator and denominator.We have
xx2 + 1
=x(1)
x2(1+ 1/x2)=
1x· 11+ 1/x2
limx→∞
xx2 + 1
= limx→∞
1x
11+ 1/x2
= limx→∞
1x· limx→∞
11+ 1/x2
= 0 · 11+ 0
= 0.
RemarkHad the higher power been in the numerator, the limit would have been∞.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 29 / 37
. . . . . .
Using the limit laws to compute limits at ∞
Example
Find limx→∞
xx2 + 1
AnswerThe limit is 0.
. .x
.y
Notice that the graph does cross the asymptote, which contradicts oneof the commonly held beliefs of what an asymptote is.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 30 / 37
. . . . . .
Solution
SolutionFactor out the largest power of x from the numerator and denominator.We have
xx2 + 1
=x(1)
x2(1+ 1/x2)=
1x· 11+ 1/x2
limx→∞
xx2 + 1
= limx→∞
1x
11+ 1/x2
= limx→∞
1x· limx→∞
11+ 1/x2
= 0 · 11+ 0
= 0.
RemarkHad the higher power been in the numerator, the limit would have been∞.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 31 / 37
. . . . . .
Another Example
Example
Find
limx→∞
2x3 + 3x+ 14x3 + 5x2 + 7
if it exists.A does not existB 1/2
C 0D ∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 32 / 37
. . . . . .
Another Example
Example
Find
limx→∞
2x3 + 3x+ 14x3 + 5x2 + 7
if it exists.A does not existB 1/2
C 0D ∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 32 / 37
. . . . . .
Solution
SolutionFactor out the largest power of x from the numerator and denominator.We have
2x3 + 3x+ 14x3 + 5x2 + 7
=x3(2+ 3/x2 + 1/x3)
x3(4+ 5/x + 7/x3)
limx→∞
2x3 + 3x+ 14x3 + 5x2 + 7
= limx→∞
2+ 3/x2 + 1/x3
4+ 5/x + 7/x3
=2+ 0+ 04+ 0+ 0
=12
Upshot
When finding limits of algebraic expressions at infinity, look at thehighest degree terms.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 33 / 37
. . . . . .
Solution
SolutionFactor out the largest power of x from the numerator and denominator.We have
2x3 + 3x+ 14x3 + 5x2 + 7
=x3(2+ 3/x2 + 1/x3)
x3(4+ 5/x + 7/x3)
limx→∞
2x3 + 3x+ 14x3 + 5x2 + 7
= limx→∞
2+ 3/x2 + 1/x3
4+ 5/x + 7/x3
=2+ 0+ 04+ 0+ 0
=12
Upshot
When finding limits of algebraic expressions at infinity, look at thehighest degree terms.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 33 / 37
. . . . . .
Still Another Example
Example
Find
limx→∞
√3x4 + 7x2 + 3
.
.√
3x4 + 7 ∼√
3x4 =√3x2
AnswerThe limit is
√3.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 34 / 37
. . . . . .
Still Another Example
Example
Find
limx→∞
√3x4 + 7x2 + 3
.
.√
3x4 + 7 ∼√
3x4 =√3x2
AnswerThe limit is
√3.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 34 / 37
. . . . . .
Solution
Solution
limx→∞
√3x4 + 7x2 + 3
= limx→∞
√x4(3+ 7/x4)
x2(1+ 3/x2)
= limx→∞
x2√
(3+ 7/x4)
x2(1+ 3/x2)
= limx→∞
√(3+ 7/x4)
1+ 3/x2
=
√3+ 01+ 0
=√3.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 35 / 37
. . . . . .
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 an expressionthat we can use the limit laws on.
limx→∞
(√4x2 + 17− 2x
)= lim
x→∞
(√4x2 + 17− 2x
)·√4x2 + 17+ 2x√4x2 + 17+ 2x
= limx→∞
(4x2 + 17)− 4x2√4x2 + 17+ 2x
= limx→∞
17√4x2 + 17+ 2x
= 0
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 36 / 37
. . . . . .
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 an expressionthat we can use the limit laws on.
limx→∞
(√4x2 + 17− 2x
)= lim
x→∞
(√4x2 + 17− 2x
)·√4x2 + 17+ 2x√4x2 + 17+ 2x
= limx→∞
(4x2 + 17)− 4x2√4x2 + 17+ 2x
= limx→∞
17√4x2 + 17+ 2x
= 0
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 36 / 37
. . . . . .
Kick it up a notch
Example
Compute limx→∞
(√4x2 + 17x− 2x
).
SolutionSame trick, different answer:
limx→∞
(√4x2 + 17x− 2x
)= lim
x→∞
(√4x2 + 17x− 2x
)·√4x2 + 17+ 2x√4x2 + 17x+ 2x
= limx→∞
(4x2 + 17x)− 4x2√4x2 + 17x+ 2x
= limx→∞
17x√4x2 + 17x+ 2x
= limx→∞
17√4+ 17/x+ 2
=174
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 37 / 37
. . . . . .
Kick it up a notch
Example
Compute limx→∞
(√4x2 + 17x− 2x
).
SolutionSame trick, different answer:
limx→∞
(√4x2 + 17x− 2x
)= lim
x→∞
(√4x2 + 17x− 2x
)·√4x2 + 17+ 2x√4x2 + 17x+ 2x
= limx→∞
(4x2 + 17x)− 4x2√4x2 + 17x+ 2x
= limx→∞
17x√4x2 + 17x+ 2x
= limx→∞
17√4+ 17/x+ 2
=174
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 37 / 37
. . . . . .
Summary
I Infinity is a more complicated concept than a single number.There are rules of thumb, but there are also exceptions.
I Take a two-pronged approach to limits involving infinity:I Look at the expression to guess the limit.I Use limit rules and algebra to verify it.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Infinity September 22, 2010 38 / 37