Lesson 6: Limits Involving Infinity (Section 41 handout)

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.


1


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.


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.


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


2


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”.


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.


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−


3


Notes
y

1
lim =∞
x→0+ x
1
lim = −∞
x→0− x x
1
lim =∞
x→0 x 2


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.


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+


4


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 +∞.”


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


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


5


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


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.


Multiplying infinite limits

∞·∞=∞
∞ · (−∞) = −∞
(−∞) · (−∞) = ∞
Fact

The product of two infinite limits is infinite.


6


Dividing by Infinity

L
=0
∞

Fact

The quotient of a finite limit by an infinite limit is zero.


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.


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!


7


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.


Indeterminate forms are like Tug Of War
Notes

Which side wins depends on which side is stronger.


Outline
Notes

Inﬁnite Limits
Vertical Asymptotes
Limit “Laws” with Inﬁnite Limits

Limits at ∞
Algebraic rates of growth


8


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!


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


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.


9


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 ∞.


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 ∞


Solution
Notes


10


Still Another Example
Notes

Example
Find √
3x 4 + 7
lim
x→∞ x2 + 3


Solution
Notes


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


11


Kick it up a notch
Notes
Example
Compute lim 4x 2 + 17x − 2x .
x→∞


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.


Notes

12

Lesson 6: Limits Involving Infinity (Section 41 handout)

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