Lesson 4: Limits Involving Infinity

Section 2.5Limits Involving Infinity

Math 1a

February 4, 2008

Announcements

I Syllabus available on course website

I All HW on website now

I No class Monday 2/18

I ALEKS due Wednesday 2/20

Outline

Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limits

Limits at InfinityAlgebraic rates of growthExponential rates of growthRationalizing to get a limit

Worksheet

Infinite Limits

DefinitionThe notation

limx→a

f (x) =∞

means that the values of f (x) can be made arbitrarily large (aslarge as we please) by taking x sufficiently close to a but not equalto a.

DefinitionThe notation

limx→a

f (x) = −∞

means that the values of f (x) can be made arbitrarily largenegative by taking x sufficiently close to a but not equal to a.

Of course we have definitions for left- and right-hand infinite limits.

Vertical Asymptotes

DefinitionThe line x = a is called a vertical asymptote of the curvey = f (x) if at 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

limx→0+

1

x=∞

limx→0−

1

x= −∞

limx→0

1

x2=∞

Finding limits at trouble spots

Example

Let

f (t) =t2 + 2

t2 − 3t + 2

Find limt→a−

f (t) and limt→a+

f (t) for each a at which f is not

continuous.

SolutionThe denominator factors as (t − 1)(t − 2). We can record thesigns of the factors on the number line.

Finding limits at trouble spots

Example

Let

f (t) =t2 + 2

t2 − 3t + 2

Find limt→a−

f (t) and limt→a+

f (t) for each a at which f is not

continuous.

SolutionThe denominator factors as (t − 1)(t − 2). We can record thesigns of the factors on the number line.

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+ ±∞ − ∓∞ +

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+ ±∞ − ∓∞ +

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+ ±∞ − ∓∞ +

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+ ±∞ − ∓∞ +

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+

±∞ − ∓∞ +

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+ ±∞

− ∓∞ +

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+ ±∞ −

∓∞ +

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+ ±∞ − ∓∞

+

(t − 1)−

1

0 +

(t − 2)−

2

0 +

(t2 + 2)+

f (t)1 2

+ ±∞ − ∓∞ +

Limit Laws with infinite limitsTo aid your intuition

I The sum of positive infinite limits is ∞. That is

∞+∞ =∞

I The sum of negative infinite limits is −∞.

−∞−∞ = −∞

I The sum of a finite limit and an infinite limit is infinite.

a +∞ =∞a−∞ = −∞

Rules of Thumb with infinite limitsDon’t try this at home!

I The sum of positive infinite limits is ∞. That is

∞+∞ =∞

I The sum of negative infinite limits is −∞.

−∞−∞ = −∞

I The sum of a finite limit and an infinite limit is infinite.

a +∞ =∞a−∞ = −∞

Rules of Thumb with infinite limitsI The product of a finite limit and an infinite limit is infinite if

the finite limit is not 0.

a · ∞ =

{∞ if a > 0

−∞ if a < 0.

a · (−∞) =

{−∞ if a > 0

∞ if a < 0.

I The product of two infinite limits is infinite.

∞ ·∞ =∞∞ · (−∞) = −∞

(−∞) · (−∞) =∞

I The quotient of a finite limit by an infinite limit is zero:

a

∞= 0.

Indeterminate Limits

I Limits of the form 0 · ∞ and ∞−∞ are indeterminate. Thereis no rule for evaluating such a form; the limit must beexamined more closely.

I Limits of the form1

0are also indeterminate.

Indeterminate Limits

I Limits of the form 0 · ∞ and ∞−∞ are indeterminate. Thereis no rule for evaluating such a form; the limit must beexamined more closely.

I Limits of the form1

0are also indeterminate.

Outline

Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limits

Limits at InfinityAlgebraic rates of growthExponential rates of growthRationalizing to get a limit

Worksheet

DefinitionLet f be a function 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 welike, by taking x sufficiently large.

DefinitionThe 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!

DefinitionLet f be a function 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 welike, by taking x sufficiently large.

DefinitionThe 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!

DefinitionLet f be a function 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 welike, by taking x sufficiently large.

DefinitionThe 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!

TheoremLet n be a positive integer. Then

I limx→∞

1

xn= 0

I limx→−∞

1

xn= 0

Using the limit laws to compute limits at ∞

Example

Find

limx→∞

2x3 + 3x + 1

4x3 + 5x2 + 7

if it exists.

A does not exist

B 1/2

C 0

D ∞

Using the limit laws to compute limits at ∞

Example

Find

limx→∞

2x3 + 3x + 1

4x3 + 5x2 + 7

if it exists.

A does not exist

B 1/2

C 0

D ∞

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

2x3 + 3x + 1

4x3 + 5x2 + 7=

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

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

limx→∞

2x3 + 3x + 1

4x3 + 5x2 + 7= lim

x→∞

2 + 3/x2 + 1/x3

4 + 5/x + 7/x3

=2 + 0 + 0

4 + 0 + 0=

1

2

Upshot

When finding limits of algebraic expressions at infinitely, look atthe highest degree terms.

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

2x3 + 3x + 1

4x3 + 5x2 + 7=

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

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

limx→∞

2x3 + 3x + 1

4x3 + 5x2 + 7= lim

x→∞

2 + 3/x2 + 1/x3

4 + 5/x + 7/x3

=2 + 0 + 0

4 + 0 + 0=

1

2

Upshot

When finding limits of algebraic expressions at infinitely, look atthe highest degree terms.

Another Example

Example

Find

limx→∞

√3x4 + 7

x2 + 3

SolutionThe limit is

√3.

Another Example

Example

Find

limx→∞

√3x4 + 7

x2 + 3

SolutionThe limit is

√3.

Example

Make a conjecture about limx→∞

x2

2x.

SolutionThe limit is zero. Exponential growth is infinitely faster thangeometric growth

Example

Make a conjecture about limx→∞

x2

2x.

SolutionThe limit is zero. Exponential growth is infinitely faster thangeometric growth

Rationalizing to get a limit

Example

Compute limx→∞

(√4x2 + 17− 2x

).

SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.

Rationalizing to get a limit

Example

Compute limx→∞

(√4x2 + 17− 2x

).

SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.

Outline

Infinite LimitsVertical AsymptotesInfinite Limits we KnowLimit “Laws” with Infinite LimitsIndeterminate Limits

Limits at InfinityAlgebraic rates of growthExponential rates of growthRationalizing to get a limit

Worksheet

Worksheet