1.
Section 1.6
Limits involving Inﬁnity
V63.0121.002.2010Su, Calculus I
New York University
May 20, 2010
Announcements
Oﬃce Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here)
Quiz 1 Thursday on 1.1–1.4
3.
Objectives
“Intuit” limits involving
inﬁnity by eyeballing the
expression.
Show limits involving inﬁnity
by algebraic manipulation
and conceptual argument.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 3 / 37
4.
Recall the deﬁnition of limit
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.
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5.
Recall the unboundedness problem
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 5 / 37
6.
Recall the unboundedness problem
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 5 / 37
7.
Recall the unboundedness problem
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 5 / 37
8.
Recall the unboundedness problem
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 5 / 37
9.
Outline
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
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 6 / 37
10.
Inﬁnite Limits
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.
x
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
11.
Inﬁnite Limits
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”.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
12.
Inﬁnite Limits
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”.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
13.
Inﬁnite Limits
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”.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
14.
Inﬁnite Limits
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”.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
15.
Inﬁnite Limits
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”.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
16.
Inﬁnite Limits
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”.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
17.
Inﬁnite Limits
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”.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 7 / 37
18.
Negative Inﬁnity
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 8 / 37
19.
Negative Inﬁnity
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 8 / 37
20.
Vertical Asymptotes
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−
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 9 / 37
21.
Inﬁnite Limits we Know
y
1
lim+ =∞
x→0 x
x
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 10 / 37
22.
Inﬁnite Limits we Know
y
1
lim+ =∞
x→0 x
1
lim = −∞
− x
x→0 x
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 10 / 37
23.
Inﬁnite Limits we Know
y
1
lim+ =∞
x→0 x
1
lim = −∞
− x
x→0 x
1
lim 2 = ∞
x→0 x
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 10 / 37
24.
Finding limits at trouble spots
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
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 11 / 37
25.
Finding limits at trouble spots
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 11 / 37
27.
Use the number line
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
So
28.
Use the number line
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
+
(x 2 + 2)
So
29.
Use the number line
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
+
(x 2 + 2)
f (x)
1 2
So
30.
Use the number line
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
+
(x 2 + 2)
+
f (x)
1 2
So
31.
Use the number line
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
+
(x 2 + 2)
+ +∞
f (x)
1 2
So
lim f (x) = +∞
x→1−
32.
Use the number line
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
+
(x 2 + 2)
+ +∞ −∞
f (x)
1 2
So
lim f (x) = +∞
x→1−
lim f (x) = −∞
x→1+
33.
Use the number line
− 0 +
(x − 1)
1
− 0 +
(x − 2)
2
+
(x 2 + 2)
+ +∞ −∞ −
f (x)
1 2
So
lim f (x) = +∞
x→1−
lim f (x) = −∞
x→1+
34.
Use the number line
− 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) = −∞
x→1+
35.
Use the number line
− 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+
36.
Use the number line
− 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+
37.
Use the number line
− 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+
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 12 / 37
38.
In English, now
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 +∞.”
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 13 / 37
39.
The graph so far
y
x
−1 1 2 3
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
40.
The graph so far
y
x
−1 1 2 3
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
41.
The graph so far
y
x
−1 1 2 3
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
42.
The graph so far
y
x
−1 1 2 3
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
43.
The graph so far
y
x
−1 1 2 3
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 14 / 37
44.
Limit Laws (?) with inﬁnite limits
If lim f (x) = ∞ and lim g (x) = ∞, then lim (f (x) + g (x)) = ∞.
x→a x→a x→a
That is,
∞+∞=∞
If lim f (x) = −∞ and lim g (x) = −∞, then
x→a x→a
lim (f (x) + g (x)) = −∞. That is,
x→a
−∞ − ∞ = −∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 15 / 37
45.
Rules of Thumb with inﬁnite limits
If lim f (x) = ∞ and lim g (x) = ∞, then lim (f (x) + g (x)) = ∞.
x→a x→a x→a
That is,
∞+∞=∞
If lim f (x) = −∞ and lim g (x) = −∞, then
x→a x→a
lim (f (x) + g (x)) = −∞. That is,
x→a
−∞ − ∞ = −∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 15 / 37
46.
Rules of Thumb with inﬁnite limits
If lim f (x) = L and lim g (x) = ±∞, then lim (f (x) + g (x)) = ±∞.
x→a x→a x→a
That is,
L+∞=∞
L − ∞ = −∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 16 / 37
47.
Rules of Thumb with inﬁnite limits
Kids, don’t try this at home!
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.
−∞ if L > 0
L · (−∞) =
∞ if L < 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 17 / 37
48.
Multiplying inﬁnite limits
Kids, don’t try this at home!
The product of two inﬁnite limits is inﬁnite.
∞·∞=∞
∞ · (−∞) = −∞
(−∞) · (−∞) = ∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 18 / 37
49.
Dividing by Inﬁnity
Kids, don’t try this at home!
The quotient of a ﬁnite limit by an inﬁnite limit is zero:
L
=0
∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 19 / 37
50.
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 20 / 37
51.
Indeterminate Limit forms
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 sin(1/x)
x→0 0
in the left- or right-hand sense. There are inﬁnitely many vertical
asymptotes arbitrarily close to 0!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 21 / 37
52.
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 22 / 37
53.
Indeterminate forms are like Tug Of War
Which side wins depends on which side is stronger.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 23 / 37
54.
Outline
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
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 24 / 37
55.
Limits at inﬁnity
Deﬁnition
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 25 / 37
56.
Limits at inﬁnity
Deﬁnition
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→−∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 25 / 37
57.
Limits at inﬁnity
Deﬁnition
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!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 25 / 37
58.
Basic limits at inﬁnity
Theorem
Let n be a positive integer. Then
1
lim =0
x→∞ x n
1
lim =0
x→−∞ x n
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 26 / 37
59.
Using the limit laws to compute limits at ∞
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 ∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 27 / 37
60.
Using the limit laws to compute limits at ∞
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 ∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 27 / 37
61.
Solution
Factor out the largest power of x from the numerator and denominator.
We have
2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 )
= 3
4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 )
2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3
lim 3 + 5x 2 + 7
= lim
x→∞ 4x x→∞ 4 + 5/x + 7/x 3
2+0+0 1
= =
4+0+0 2
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 28 / 37
62.
Solution
Factor out the largest power of x from the numerator and denominator.
We have
2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 )
= 3
4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 )
2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3
lim 3 + 5x 2 + 7
= lim
x→∞ 4x x→∞ 4 + 5/x + 7/x 3
2+0+0 1
= =
4+0+0 2
Upshot
When ﬁnding limits of algebraic expressions at inﬁnity, look at the highest
degree terms.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 28 / 37
63.
Another Example
Example
x
Find lim
x→∞ x2 + 1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 29 / 37
64.
Another Example
Example
x
Find lim
x→∞ x2 + 1
Answer
The limit is 0.
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65.
Solution
Again, 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 2 = lim = lim · lim
x→∞ x + 1 x→∞ x 1 + 1/x 2 x→∞ x x→∞ 1 + 1/x 2
1
=0· = 0.
1+0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 30 / 37
66.
Another Example
Example
x
Find lim
x→∞ x2 + 1
Answer
The limit is 0.
y
x
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67.
Another Example
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 31 / 37
68.
Solution
Again, 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 2 = lim = lim · lim
x→∞ x + 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 ∞.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 32 / 37
69.
Another Example
Example
Find
√
3x 4 + 7
lim
x→∞ x2 + 3
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 33 / 37
70.
Another Example
Example
Find
√ √
3x 4 + 7 ∼ 3x 4 = 3x 2
√
3x 4 + 7
lim
x→∞ x2 + 3
Answer
√
The limit is 3.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 33 / 37
72.
Rationalizing to get a limit
Example
Compute lim 4x 2 + 17 − 2x .
x→∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 35 / 37
73.
Rationalizing to get a limit
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
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 35 / 37
74.
Kick it up a notch
Example
Compute lim 4x 2 + 17x − 2x .
x→∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 36 / 37
76.
Summary
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.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.6 Limits involving Inﬁnity May 20, 2010 37 / 37
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