1.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Notes
Section 1.6
Limits involving Inﬁnity
V63.0121.041, Calculus I
New York University
September 21, 2010
Announcements
Announcements
Notes
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 2 / 35
Objectives
Notes
“Intuit” limits involving
inﬁnity by eyeballing the
expression.
Show limits involving inﬁnity
by algebraic manipulation
and conceptual argument.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 3 / 35
1
2.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Recall the deﬁnition of limit
Notes
Deﬁnition
We write
lim f (x) = L
x→a
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 as we
like) by taking x to be suﬃciently close to a (on either side of a) but not
equal to a.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 4 / 35
Recall the unboundedness problem
Notes
1
Recall why lim+ doesn’t exist.
x→0 x
y
L?
x
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 21, 2010 5 / 35
Outline
Notes
Inﬁnite Limits
Vertical Asymptotes
Inﬁnite Limits we Know
Limit “Laws” with Inﬁnite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 6 / 35
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3.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Inﬁnite Limits
Notes
Deﬁnition
The notation y
lim f (x) = ∞
x→a
means that values of f (x) can be
made arbitrarily large (as large as
we please) by taking x suﬃciently
close to a but not equal to a.
“Large” takes the place of x
“close to L”.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 7 / 35
Negative Inﬁnity
Notes
Deﬁnition
The notation
lim f (x) = −∞
x→a
means that the values of f (x) can be made arbitrarily large negative (as
large as we please) by taking x suﬃciently close to a but not equal to a.
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 21, 2010 8 / 35
Vertical Asymptotes
Notes
Deﬁnition
The line x = a is called a vertical asymptote of the curve y = f (x) if at
least one of the following is true:
lim f (x) = ∞ lim f (x) = −∞
x→a x→a
lim+ f (x) = ∞ lim f (x) = −∞
x→a x→a+
lim f (x) = ∞ lim f (x) = −∞
x→a− x→a−
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 9 / 35
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4.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Inﬁnite Limits we Know
Notes
y
1
lim =∞
x→0+ x
1
lim = −∞
x→0− x x
1
lim =∞
x→0 x 2
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 10 / 35
Finding limits at trouble spots
Notes
Example
Let
x2 + 2
f (x) =
x 2 − 3x + 2
Find lim f (x) and lim+ f (x) for each a at which f is not continuous.
x→a− x→a
Solution
The denominator factors as (x − 1)(x − 2). We can record the signs of the
factors on the number line.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 11 / 35
Use the number line
Notes
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
+
(x 2 + 2)
+ +∞ −∞ − −∞ +∞ +
f (x)
1 2
So
lim f (x) = +∞ lim f (x) = −∞
x→1− x→2−
lim f (x) = −∞ lim f (x) = +∞
x→1+ x→2+
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 12 / 35
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5.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
In English, now
Notes
To explain the limit, you can say:
“As x → 1− , the numerator approaches 3, and the denominator
approaches 0 while remaining positive. So the limit is +∞.”
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 13 / 35
The graph so far
Notes
lim f (x) = + ∞ lim f (x) = − ∞
x→1− x→2−
lim f (x) = − ∞ lim f (x) = + ∞
x→1+ x→2+
y
x
−1 1 2 3
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 14 / 35
Rules of Thumb with inﬁnite limits
Notes
Fact ∞+∞=∞
If lim f (x) = ∞ and
x→a
lim g (x) = ∞, then
x→a
−∞ − ∞ = ∞
lim (f (x) + g (x)) = ∞.
x→a
If lim f (x) = −∞ and
x→a
lim g (x) = −∞, then
x→a
lim (f (x) + g (x)) = −∞.
x→a
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 15 / 35
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6.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Rules of Thumb with inﬁnite limits
Notes
L+∞=∞
L − ∞ = −∞
Fact
If lim f (x) = L and lim g (x) = ±∞, then
x→a x→a
lim (f (x) + g (x)) = ±∞.
x→a
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 16 / 35
Rules of Thumb with inﬁnite limits
Kids, don’t try this at home! Notes
∞ if L > 0
L·∞=
−∞ if L < 0.
Fact
The product of a ﬁnite limit and an inﬁnite limit is inﬁnite if the ﬁnite
limit is not 0.
−∞ if L > 0
L · (−∞) =
∞ if L < 0.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 17 / 35
Multiplying inﬁnite limits
Kids, don’t try this at home! Notes
∞·∞=∞
∞ · (−∞) = −∞
(−∞) · (−∞) = ∞
Fact
The product of two inﬁnite limits is inﬁnite.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 18 / 35
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7.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Dividing by Inﬁnity
Kids, don’t try this at home! Notes
L
=0
∞
Fact
The quotient of a ﬁnite limit by an inﬁnite limit is zero.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 19 / 35
Dividing by zero is still not allowed
Notes
1
=∞
0
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 21, 2010 20 / 35
Indeterminate Limit forms
Notes
L
Limits of the form are indeterminate. There is no rule for evaluating
0
such a form; the limit must be examined more closely. Consider these:
1 −1
lim =∞ lim = −∞
x→0 x2 x→0 x2
1 1
lim =∞ lim = −∞
x→0+ x x→0− x
1 L
Worst, lim is of the form , but the limit does not exist, even
x→0x sin(1/x) 0
in the left- or right-hand sense. There are inﬁnitely many vertical
asymptotes arbitrarily close to 0!
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 21 / 35
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8.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Indeterminate Limit forms
Notes
Limits of the form 0 · ∞ and ∞ − ∞ are also indeterminate.
Example
1
The limit lim+ sin x · is of the form 0 · ∞, but the answer is 1.
x→0 x
1
The limit lim+ sin2 x · is of the form 0 · ∞, but the answer is 0.
x→0 x
1
The limit lim+ sin x · 2 is of the form 0 · ∞, but the answer is ∞.
x→0 x
Limits of indeterminate forms may or may not “exist.” It will depend on
the context.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 22 / 35
Indeterminate forms are like Tug Of War
Notes
Which side wins depends on which side is stronger.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 23 / 35
Outline
Notes
Inﬁnite Limits
Vertical Asymptotes
Inﬁnite Limits we Know
Limit “Laws” with Inﬁnite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Rationalizing to get a limit
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 24 / 35
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9.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Deﬁnition Notes
Let f be a function deﬁned on some interval (a, ∞). Then
lim f (x) = L
x→∞
means that the values of f (x) can be made as close to L as we like, by
taking x suﬃciently large.
Deﬁnition
The line y = L is a called a horizontal asymptote of the curve y = f (x)
if either
lim f (x) = L or lim f (x) = L.
x→∞ x→−∞
y = L is a horizontal line!
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 25 / 35
Basic limits at inﬁnity
Notes
Theorem
Let n be a positive integer. Then
1
lim =0
x→∞ x n
1
lim =0
x→−∞ x n
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 26 / 35
Using the limit laws to compute limits at ∞
Notes
Example
x
Find lim
x→∞ x2 + 1
Answer
The limit is 0.
y
x
Notice that the graph does cross the asymptote, which contradicts one of
the heuristic deﬁnitions of asymptote.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 27 / 35
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10.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Solution
Notes
Solution
Factor out the largest power of x from the numerator and denominator.
We have
x x(1) 1 1
= 2 = ·
x2 + 1 x (1 + 1/x 2 ) x 1 + 1/x 2
x 1 1 1 1
lim = lim = lim · lim
x→∞ x 2 + 1 x→∞ x 1 + 1/x 2 x→∞ x x→∞ 1 + 1/x 2
1
=0· = 0.
1+0
Remark
Had 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 21, 2010 28 / 35
Another Example
Notes
Example
Find
2x 3 + 3x + 1
lim
x→∞ 4x 3 + 5x 2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 29 / 35
Solution
Notes
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 30 / 35
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11.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Still Another Example
Notes
Example
Find √
3x 4 + 7
lim
x→∞ x2 + 3
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 31 / 35
Solution
Notes
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 32 / 35
Rationalizing to get a limit
Notes
Example
Compute lim 4x 2 + 17 − 2x .
x→∞
Solution
This limit is of the form ∞ − ∞, which we cannot use. So we rationalize
the numerator (the denominator is 1) to get an expression that we can use
the limit laws on.
√
4x 2 + 17 + 2x
lim 4x 2 + 17 − 2x = lim 4x 2 + 17 − 2x · √
x→∞ x→∞ 4x 2 + 17 + 2x
(4x 2 + 17) − 4x 2
= lim √
x→∞ 4x 2 + 17 + 2x
17
= lim √ =0
x→∞ 4x 2 + 17 + 2x
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 33 / 35
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12.
V63.0121.041, Calculus I Section 1.6 : Limits involving Inﬁnity September 21, 2010
Kick it up a notch
Notes
Example
Compute lim 4x 2 + 17x − 2x .
x→∞
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 34 / 35
Summary
Notes
Inﬁnity is a more complicated concept than a single number. There
are rules of thumb, but there are also exceptions.
Take a two-pronged approach to limits involving inﬁnity:
Look at the expression to guess the limit.
Use limit rules and algebra to verify it.
V63.0121.041, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity September 21, 2010 35 / 35
Notes
12
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