Infinity is a complicated concept, but there are rules for dealing with both limits at infinity and infinite limits.

1. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Sec on 1.5
Limits Involving Infinity
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
February 9, 2011
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Notes
Announcements
Get-to-know-you extra
credit due Friday
February 11
Quiz 1 is next week in
recita on. Covers
Sec ons 1.1–1.4
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Notes
Objectives
“Intuit” limits involving infinity by
eyeballing the expression.
Show limits involving infinity by
algebraic manipula on and conceptual
argument.
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2. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Recall the definition of limit
Defini on
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.
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Notes
The unboundedness problem
y
1
Recall why lim+ doesn’t
x→0 x
exist.
No ma er how thin we draw
the strip to the right of x = 0, L?
we cannot “capture” the graph
inside the box.
. x
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Notes
Infinite Limits
Defini on
y
The nota on
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
. x
to a but not equal to a.
“Large” takes the place of “close to L”.
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3. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Negative Infinity
Defini on
The nota on
lim f(x) = −∞
x→a
means that the values of f(x) can be made arbitrarily large nega ve
(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 (nega ve) number.
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Notes
Vertical Asymptotes
Defini on
The line x = a is called a ver cal 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−
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Notes
Infinite Limits we Know
y
1
lim =∞
x→0+ x
1
lim = −∞
x→0− x . x
1
lim = ∞
x→0 x2
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4. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Finding limits at trouble spots
Example
Let
x2 + 2
f(x) =
x2 − 3x + 2
Find lim− f(x) and lim+ f(x) for each a at which f is not con nuous.
x→a x→a
Solu on
The denominator factors as (x − 1)(x − 2). We can record the signs
of the factors on the number line.
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Notes
Use the number line
small
small
−. 0 +
(x − 1)
small
small
1
− 0 +
(x − 2)
2
+
(x2 + 2)
+ +∞ −∞−−∞ +∞ +
f(x)
1 2
lim f(x) = + ∞ lim f(x) = − ∞
x→1− x→2−
lim f(x) = − ∞ lim f(x) = + ∞
x→1+ x→2+
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Notes
In English, now
To explain the limit, you can say:
“As x → 1− , the numerator approaches 3, and the denominator
approaches 0 while remaining posi ve. So the limit is +∞.”
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5. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
The graph so far
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
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Notes
Rules of Thumb with infinite limits
Fact
The sum of two posi ve or two nega ve infinite limits is infinite.
If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞.
x→a x→a x→a
If lim f(x) = −∞ and lim g(x) = −∞, then
x→a x→a
lim (f(x) + g(x)) = −∞.
x→a
Remark
We don’t say anything here about limits of the form ∞ − ∞.
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Notes
Rules of Thumb with infinite limits
Fact
The sum of a finite limit and an infinite limit is infinite.
If lim f(x) = L and lim g(x) = ±∞, then
x→a x→a
lim (f(x) + g(x)) = ±∞.
x→a
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6. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Rules of Thumb with infinite limits
Fact
The product of a finite limit and an infinite limit is infinite if the finite
limit is not 0.
If lim f(x) = L, lim g(x) = ∞, and L > 0, then
x→a x→a
lim f(x) · g(x) = ∞.
x→a
If lim f(x) = L, lim g(x) = ∞, and L < 0, then
x→a x→a
lim f(x) · g(x) = −∞.
x→a
If lim f(x) = L, lim g(x) = −∞, and L > 0, then
x→a x→a
lim f(x) · g(x) = −∞.
x→a
. If lim f(x) = L, lim g(x) = −∞, and L < 0, then
x→a x→a
lim f(x) · g(x) = ∞.
x→a .
Notes
Multiplying infinite limits
Fact
The product of two infinite limits is infinite.
If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞.
x→a x→a x→a
If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞.
x→a x→a x→a
If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞.
x→a x→a x→a
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Notes
Dividing by Infinity
Fact
The quo ent of a finite limit by an infinite limit is zero.
f(x)
If lim f(x) = L and lim g(x) = ±∞, then lim = 0.
x→a x→a x→a g(x)
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7. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Dividing by zero is still not allowed
1=∞
.
0
There are examples of such limit forms where the limit is ∞, −∞,
undecided between the two, or truly neither.
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Notes
Indeterminate Limit forms
L
Limits of the form are indeterminate. There is no rule for
0
evalua ng 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,
x→0 x sin(1/x) 0
even in the le - or right-hand sense. There are infinitely many
. ver cal asymptotes arbitrarily close to 0!
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Notes
Indeterminate Limit forms
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.
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8. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Indeterminate forms are like Tug Of War
Which side wins depends on which side is stronger.
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Notes
Outline
Ver cal Asymptotes
Infinite Limits we Know
Limit “Laws” with Infinite Limits
Indeterminate Limit forms
Limits at ∞
Algebraic rates of growth
Ra onalizing to get a limit
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Notes
Limits at Infinity
Defini on
Let f be a func on 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.
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9. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Horizontal Asymptotes
Defini on
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!
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Notes
Basic limits at infinity
Theorem
Let n be a posi ve integer. Then
1
lim =0
x→∞ xn
1
lim =0
x→−∞ xn
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Notes
Limit laws at infinity
Fact
Any limit law that concerns finite limits at a finite point a is s ll true
if the finite point is replaced by ±∞.
That is, if lim f(x) = L and lim g(x) = M, then
x→∞ x→∞
lim (f(x) + g(x)) = L + M
x→∞
lim (f(x) − g(x)) = L − M
x→∞
lim cf(x) = c · L (for any constant c)
x→∞
lim f(x) · g(x) = L · M
x→∞
f(x) L
lim = (if M ̸= 0), etc.
. x→∞ g(x) M
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10. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Using the limit laws to compute Notes
limits at ∞
Example
x
Find lim
x→∞ x2 +1
Answer
The limit is 0. No ce that
the graph does cross the
y
asymptote, which
contradicts one of the . x
commonly held beliefs of
what an asymptote is.
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Notes
Solution
Solu on
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Notes
Another Example
Example
Find
2x3 + 3x + 1
lim
x→∞ 4x3 + 5x2 + 7
if it exists.
A does not exist
B 1/2
C 0
D ∞
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11. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Solution
Solu on
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Notes
Still Another Example
Example
Find √
3x4 + 7
lim
x→∞ x2 + 3
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Notes
Solution
Solu on
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12. . V63.0121.001: Calculus I
. Sec on 1.5: Limits Involving Infinity
. February 9, 2011
Notes
Rationalizing to get a limit
.
Example
(√ )
Compute lim 4x2 + 17 − 2x .
x→∞
Solu on
This limit is of the form ∞ − ∞, which we cannot use. So we ra onalize the
numerator (the denominator is 1) to get an expression that we can use the
limit laws on.
(√ ) (√ ) √4x2 + 17 + 2x
lim 4x2 + 17 − 2x = lim 4x2 + 17 − 2x · √
x→∞ x→∞ 4x2 + 17 + 2x
(4x2 + 17) − 4x2 17
= lim √ = lim √ =0
x→∞ 4x2 + 17 + 2x x→∞ 4x2 + 17 + 2x
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Notes
Kick it up a notch
Example
(√ )
Compute lim 4x2 + 17x − 2x .
x→∞
Solu on
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Notes
Summary
Infinity is a more complicated concept than a single number.
There are rules of thumb, but there are also excep ons.
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.
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