lesson 6: limits involving infinity (slides)

..

Sec on 1.6Limits Involving Infinity

V63.0121.011: Calculus IProfessor Ma hew Leingang

New York University

February 9, 2011

Announcements

I Get-to-know-you extracredit due FridayFebruary 11

I Quiz 1 is next week inrecita on. CoversSec ons 1.1–1.4

Objectives

I “Intuit” limits involving infinity byeyeballing the expression.

I Show limits involving infinity byalgebraic manipula on and conceptualargument.

Recall the definition of limitDefini onWe 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 Las we like) by taking x to be sufficiently close to a (on either side ofa) but not equal to a.

The unboundedness problem

Recall why limx→0+

1xdoesn’t

exist.No ma er how thin we drawthe strip to the right of x = 0,we cannot “capture” the graphinside the box.

.. x.

y

..

L?

OutlineInfinite Limits

Ver cal AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limit forms

Limits at∞Algebraic rates of growthRa onalizing to get a limit

Infinite LimitsDefini onThe nota on

limx→a

f(x) = ∞

means that values of f(x) can bemade arbitrarily large (as large as weplease) by taking x sufficiently closeto a but not equal to a. .. x.

y

I “Large” takes the place of “close to L”.

Negative InfinityDefini onThe nota on

limx→a

f(x) = −∞

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 notequal to a.

I We call a number large or small based on its absolute value. So−1, 000, 000 is a large (nega ve) number.

Vertical Asymptotes

Defini onThe line x = a is called a ver cal asymptote of the curve y = f(x) ifat least one of the following is true:

I limx→a

f(x) = ∞I lim

x→a+f(x) = ∞

I limx→a−

f(x) = ∞

I limx→a

f(x) = −∞I lim

x→a+f(x) = −∞

I limx→a−

f(x) = −∞

Infinite Limits we Know

I limx→0+

1x= ∞

I limx→0−

1x= −∞

I limx→0

1x2

= ∞

.. x.

y

............

Finding limits at trouble spotsExample

Letf(x) =

x2 + 2x2 − 3x+ 2

Find limx→a−

f(x) and limx→a+

f(x) for each a at which f is not con nuous.

Solu onThe denominator factors as (x− 1)(x− 2). We can record the signsof the factors on the number line.

Use the number line

.. (x− 1)

.−.

small

.

small

..1

. 0. +.

(x− 2)

.−..

2

.

0

.

small

.

small

.+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞ limx→2−

f(x) = −∞

limx→1+

f(x) = −∞ limx→2+

f(x) = +∞

Use the number line

.. (x− 1).−.

small

.

small

..1

. 0. +

.

(x− 2)

.−..

2

.

0

.

small

.

small

.+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞ limx→2−

f(x) = −∞

limx→1+

f(x) = −∞ limx→2+

f(x) = +∞

Use the number line

.. (x− 1).−.

small

.

small

..1

. 0. +.

(x− 2)

.−..

2

.

0

.

small

.

small

.+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞ limx→2−

f(x) = −∞

limx→1+

f(x) = −∞ limx→2+

f(x) = +∞

Use the number line

.. (x− 1).−. small

.

small

..1

. 0. +.

(x− 2)

.−..

2

.

0

.

small

.

small

.+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞

limx→2−

f(x) = −∞

limx→1+

f(x) = −∞ limx→2+

f(x) = +∞

Use the number line

.. (x− 1).−.

small

. small

..1

. 0. +.

(x− 2)

.−..

2

.

0

.

small

.

small

.+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞

limx→2−

f(x) = −∞

limx→1+

f(x) = −∞

limx→2+

f(x) = +∞

Use the number line

.. (x− 1).−.

small

.

small

..1

. 0. +.

(x− 2)

.−..

2

.

0

.

small

.

small

.+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞

limx→2−

f(x) = −∞

limx→1+

f(x) = −∞

limx→2+

f(x) = +∞

Use the number line

.. (x− 1).−.

small

.

small

..1

. 0. +.

(x− 2)

.−..

2

.

0

.

small.

small

.+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞ limx→2−

f(x) = −∞

limx→1+

f(x) = −∞

limx→2+

f(x) = +∞

Use the number line

.. (x− 1).−.

small

.

small

..1

. 0. +.

(x− 2)

.−..

2

.

0

.

small

.

small.

+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞ limx→2−

f(x) = −∞

limx→1+

f(x) = −∞ limx→2+

f(x) = +∞

Use the number line

.. (x− 1).−.

small

.

small

..1

. 0. +.

(x− 2)

.−..

2

.

0

.

small

.

small

.+

.

(x2 + 2)

.

+

.

f(x)

..

1

..

2

.

+

.

+∞

.

−∞

.

−

.

−∞

.

+∞

.

+

limx→1−

f(x) = +∞ limx→2−

f(x) = −∞

limx→1+

f(x) = −∞ limx→2+

f(x) = +∞

In English, now

To explain the limit, you cansay/write:

“As x → 1−, the numeratorapproaches 3, and thedenominator approaches 0while remaining posi ve. Sothe limit is+∞.”

The graph so farlimx→1−

f(x) = +∞ limx→2−

f(x) = −∞

limx→1+

f(x) = −∞ limx→2+

f(x) = +∞

.. x.

y

..−1

..1

..2

..3

Limit Laws (?) with infinite limitsFactThe sum of two posi ve or two nega ve infinite limits is infinite.

I If limx→a

f(x) = ∞ and limx→a

g(x) = ∞, then limx→a

(f(x) + g(x)) = ∞.

..

∞+∞ = ∞

I If limx→a

f(x) = −∞ and limx→a

g(x) = −∞, thenlimx→a

(f(x) + g(x)) = −∞.

..

−∞+ (−∞) = −∞

RemarkWe don’t say anything here about limits of the form∞−∞.

Rules of Thumb with infinite limitsFactThe sum of two posi ve or two nega ve infinite limits is infinite.

I If limx→a



(f(x) + g(x)) = ∞...

∞+∞ = ∞

I If limx→a



(f(x) + g(x)) = −∞.

..

−∞+ (−∞) = −∞


Rules of Thumb with infinite limitsFactThe sum of two posi ve or two nega ve infinite limits is infinite.

I If limx→a



(f(x) + g(x)) = ∞...

∞+∞ = ∞

I If limx→a



(f(x) + g(x)) = −∞. ..

−∞+ (−∞) = −∞


Rules of Thumb with infinite limitsKids, don’t try this at home!

FactThe sum of a finite limit and an infinite limit is infinite.

I If limx→a

f(x) = L and limx→a

g(x) = ±∞,

..

L+∞ = ∞L−∞ = −∞

thenlimx→a

(f(x) + g(x)) = ±∞.


FactThe sum of a finite limit and an infinite limit is infinite.

I If limx→a


g(x) = ±∞,..

L+∞ = ∞L−∞ = −∞

thenlimx→a

(f(x) + g(x)) = ±∞.


FactThe product of a finite limit and an infinite limit is infinite if the finitelimit is not 0.

I If limx→a

f(x) = L, limx→a

g(x) = ∞, and L > 0, thenlimx→a

f(x) · g(x) = ∞.

I If limx→a

f(x) = L, limx→a

g(x) = ∞, and L < 0, thenlimx→a

f(x) · g(x) = −∞.

..

L · ∞ =

{∞ if L > 0−∞ if L < 0.



I If limx→a

f(x) = L, limx→a

g(x) = ∞, and L > 0, thenlimx→a

f(x) · g(x) = ∞.

I If limx→a

f(x) = L, limx→a

g(x) = ∞, and L < 0, thenlimx→a

f(x) · g(x) = −∞. ..

L · ∞ =

{∞ if L > 0−∞ if L < 0.



I If limx→a

f(x) = L, limx→a

g(x) = −∞, and L > 0, thenlimx→a

f(x) · g(x) = −∞.

I If limx→a

f(x) = L, limx→a

g(x) = −∞, and L < 0, thenlimx→a

f(x) · g(x) = ∞.

..

L · (−∞) =

{−∞ if L > 0∞ if L < 0.



I If limx→a

f(x) = L, limx→a

g(x) = −∞, and L > 0, thenlimx→a

f(x) · g(x) = −∞.

I If limx→a

f(x) = L, limx→a

g(x) = −∞, and L < 0, thenlimx→a

f(x) · g(x) = ∞. ..

L · (−∞) =

{−∞ if L > 0∞ if L < 0.

Multiplying infinite limitsKids, don’t try this at home!

FactThe product of two infinite limits is infinite.

..

∞ ·∞ = ∞∞ · (−∞) = −∞

(−∞) · (−∞) = ∞

I If limx→a



f(x) · g(x) = ∞.

I If limx→a


g(x) = −∞, then limx→a

f(x) · g(x) = −∞.

I If limx→a



f(x) · g(x) = ∞.

Multiplying infinite limitsKids, don’t try this at home!

FactThe product of two infinite limits is infinite. ..

∞ ·∞ = ∞∞ · (−∞) = −∞

(−∞) · (−∞) = ∞

I If limx→a



f(x) · g(x) = ∞.

I If limx→a



f(x) · g(x) = −∞.

I If limx→a



f(x) · g(x) = ∞.

Dividing by InfinityKids, don’t try this at home!

FactThe quo ent of a finite limit by an infinite limit is zero.

I If limx→a


g(x) = ±∞, then limx→a

f(x)g(x)

= 0.

..

L∞

= 0

Dividing by InfinityKids, don’t try this at home!

FactThe quo ent of a finite limit by an infinite limit is zero.

I If limx→a


g(x) = ±∞, then limx→a

f(x)g(x)

= 0. ..

L∞

= 0

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.

Indeterminate Limit formsLimits of the form

L0are indeterminate. There is no rule for

evalua ng 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 le - or right-hand sense. There are infinitely manyver cal asymptotes arbitrarily close to 0!

More Indeterminate Limit formsLimits 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.


sin2 x · 1xis of the form 0 · ∞, but the answer is 0.


sin x · 1x2

is of the form 0 ·∞, but the answer is∞.

Limits of indeterminate forms may or may not “exist.” It will dependon the context.

Indeterminate forms are like Tug Of War

Which side wins depends on which side is stronger.

OutlineInfinite Limits

Ver cal AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limit forms

Limits at∞Algebraic rates of growthRa onalizing to get a limit

Limits at Infinity

Defini onLet f be a func on 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.

Horizontal Asymptotes

Defini onThe line y = L is a called a horizontal asymptote of the curvey = f(x) if either

limx→∞

f(x) = L or limx→−∞

f(x) = L.

y = L is a horizontal line!

Basic limits at infinity

TheoremLet n be a posi ve integer. Then

I limx→∞

1xn

= 0

I limx→−∞

1xn

= 0

Limit laws at infinityFactAny limit law that concerns finite limits at a finite point a is s ll trueif the finite point is replaced by±∞. That is, if lim

x→∞f(x) = L and

limx→∞

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 anyconstant c)

I limx→∞

f(x) · g(x) = L ·M

I limx→∞

f(x)g(x)

=LM

(if M ̸= 0)

I etc.

Computing limits at ∞With the limit laws

Example

Find limx→∞

xx2 + 1

if it exists.

AnswerThe limit is 0.

No ce thatthe graph does cross theasymptote, whichcontradicts one of thecommonly held beliefs ofwhat an asymptote is.

..x

.

y

Computing limits at ∞With the limit laws

Example

Find limx→∞

xx2 + 1

if it exists.

AnswerThe limit is 0. No ce thatthe graph does cross theasymptote, whichcontradicts one of thecommonly held beliefs ofwhat an asymptote is.

..x

.

y

SolutionSolu onFactor out the largest power of x from the numerator anddenominator. We have

xx2 + 1

=x(1)

x2(1+ 1/x2)=

1x· 11+ 1/x2

, so

limx→∞

xx2 + 1

= limx→∞

1x

11+ 1/x2

= limx→∞

1x· limx→∞

11+ 1/x2

= 0 · 11+ 0

= 0.

Another Example

Example

Find limx→∞

x3 + 2x2 + 43x2 + 1

if it exists.

AnswerThe limit is∞.


x3 + 2x2 + 43x2 + 1

=x3(1+ 2/x + 4/x3)

x2(3+ 1/x2)= x · 1+

2/x + 4/x3

3+ 1/x2, so

limx→∞

x3 + 2x2 + 43x2 + 1

= limx→∞

(x · 1+

2/x + 4/x3

3+ 1/x2

)=

(limx→∞

x)· 13= ∞

Yet Another ExampleExample

Findlimx→∞

2x3 + 3x+ 14x3 + 5x2 + 7

if it exists.A does not existB 1/2

C 0D ∞


2x3 + 3x+ 14x3 + 5x2 + 7

=x3(2+ 3/x2 + 1/x3)

x3(4+ 5/x + 7/x3), so

limx→∞

2x3 + 3x+ 14x3 + 5x2 + 7

= limx→∞

2+ 3/x2 + 1/x3

4+ 5/x + 7/x3=

2+ 0+ 04+ 0+ 0

=12

Upshot of the last three examples

Upshot

When finding limits of algebraic expressions at infinity, look at thehighest degree terms.

I If the higher degree is in the numerator, the limit is±∞.I If the higher degree is in the denominator, the limit is 0.I If the degrees are the same, the limit is the ra o of thetop-degree coefficients.

Still Another Example

Example

Find

limx→∞

√3x4 + 7x2 + 3

..

√3x4 + 7 ∼

√3x4 =

√3x2

AnswerThe limit is

√3.

Solution

Solu on

limx→∞

√3x4 + 7x2 + 3

= limx→∞

√x4(3+ 7/x4)

x2(1+ 3/x2)= lim

x→∞

x2√(3+ 7/x4)

x2(1+ 3/x2)

= limx→∞

√(3+ 7/x4)

1+ 3/x2=

√3+ 01+ 0

=√3.

Rationalizing to get a limitExample

Compute limx→∞

(√4x2 + 17− 2x

).

Solu onThis limit is of the form∞−∞, which we cannot use. So wera onalize the numerator (the denominator is 1) to get anexpression that 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

Kick it up a notchExample

Compute limx→∞

(√4x2 + 17x− 2x

).

Solu onSame trick, different answer:

limx→∞

(√4x2 + 17x− 2x

)= lim

x→∞

(√4x2 + 17x− 2x

)·√4x2 + 17x+ 2x√4x2 + 17x+ 2x

= limx→∞

(4x2 + 17x)− 4x2√4x2 + 17x+ 2x

= limx→∞

17x√4x2 + 17x+ 2x

= limx→∞

17√4+ 17/x+ 2

=174

Summary

I Infinity is a more complicated concept than a single number.There are rules of thumb, but there are also excep ons.

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.

lesson 6: limits involving infinity (slides)

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