Lesson 6: Continuity II, Infinite Limits

Lesson 6Continuity, Infinite Limits

Math 1a

October 5, 2007

Announcements

I No class Monday 10/8, yes class Friday 10/12

I MQC closed Sunday, open Monday

Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.

x

f (x)

a b

f (a)

f (b)

N

c

c1 c2 c3

Using the IVT

Example

Let f (x) = x3 − x − 1. Show that there is a zero for f . Estimate itwithin 1/16.

Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007

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Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007

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Back to the Questions

True or FalseAt one point in your life you were exactly three feet tall.

True or FalseAt one point in your life your height in inches equaled your weightin pounds.

True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.

Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007

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Back to the Questions

True or FalseAt one point in your life you were exactly three feet tall.

True or FalseAt one point in your life your height in inches equaled your weightin pounds.

True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.

Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007

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Back to the Questions

True or FalseAt one point in your life you were exactly three feet tall.

True or FalseAt one point in your life your height in inches equaled your weightin pounds.

True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.

Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007

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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 (as large as we please) by taking x sufficiently close to abut not equal to a.

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

Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007

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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 limx→a+ f (x) =∞I limx→a− f (x) =∞

I limx→a f (x) = −∞I limx→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 notcontinuous.

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 notcontinuous.

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

+ ±∞ − ∓∞ +

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Limit Laws with infinite limits

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 limits

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.

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 form 10 are also indeterminate.

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 form 10 are also indeterminate.

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.

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.