I. The Limit Laws

Math131 Calculus I

I.

The Limit Laws

Notes 2.3

The Limit Laws

Assumptions: c is a constant and lim f ( x ) and lim g ( x ) exist

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Limit Law in symbols

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Limit Law in words

1

lim[ f ( x ) + g ( x )] = lim f ( x ) + lim g ( x)

The limit of a sum is equal to

the sum of the limits.

2

lim[ f ( x ) ? g ( x )] = lim f ( x ) ? lim g ( x )

The limit of a difference is equal to

the difference of the limits.

3

lim cf ( x) = c lim f ( x)

The limit of a constant times a function is equal

to the constant times the limit of the function.

4

lim[ f ( x ) g ( x )] = lim f ( x ) ? lim g ( x)]

The limit of a product is equal to

the product of the limits.

5

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x¡úa

x¡úa

x¡úa

x¡úa

x¡úa

x¡úa

x¡úa

x¡úa

lim

x¡úa

x¡úa

x¡úa

f ( x)

f ( x ) lim

= x¡úa

g ( x ) lim g ( x)

(if lim g ( x) ¡Ù 0)

x¡úa

The limit of a quotient is equal to

the quotient of the limits.

x¡úa

6

lim[ f ( x)] n = [lim f ( x )] n

where n is a positive integer

7

lim c = c

The limit of a constant function is equal

to the constant.

8

lim x = a

The limit of a linear function is equal

to the number x is approaching.

9

lim x n = a n

where n is a positive integer

10

lim n x = n a

where n is a positive integer & if n is even,

we assume that a > 0

11

lim n f ( x ) = n lim f ( x )

where n is a positive integer & if n is even,

we assume that lim f ( x ) > 0

x¡úa

x¡úa

x¡úa

x¡úa

x¡úa

x¡úa

x¡úa

x¡úa

x¡ú a

Direct Substitution Property:

If f is a polynomial or rational function and a is in the domain of f,

then lim f ( x ) =

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¡°Simpler Function Property¡±:

If f ( x) = g ( x) when x ¡Ù a then lim f ( x ) = lim g ( x ) , as long as the

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limit exists.

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Math131 Calculus I

ex#1

Notes 2.3

Given lim f ( x) = 2 , lim g ( x) = ?1 , lim h( x) = 3 use the Limit Laws find lim f ( x )h( x ) ? x 2 g ( x)

x ¡ú3

ex#2

x ¡ú3

x ¡ú3

2

Evaluate lim 22 x + 1 , if it exists, by using the Limit Laws.

x¡ú2

ex#3

page 2

Evaluate:

x + 6x ? 4

lim 2 x 2 + 3 x ? 5

x ¡ú1

ex#4

Evaluate:

1 ? (1 ? x) 2

lim

x¡ú0

x

ex#5

Evaluate:

lim

h ¡ú0

h+4 ?2

h

x ¡ú3

Math131 Calculus I

Notes 2.3

page 3

Two Interesting Functions

1.

Absolute Value Function

Definition: x = ?? x if x ¡Ý 0

?? x if x < 0

Geometrically:

The absolute value of a number indicates its distance from another number.

x ? c = a means the number x is exactly _____ units away from the number _____.

x ? c < a means:

The number x is within _____ units of the number _____.

How to solve equations and inequalities involving absolute value:

Solve: |3x + 2| = 7

Solve: |x - 5| < 2

What does |x - 5| < 2 mean geometrically?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

2.

The Greatest Integer Function

Definition:

[[x]] = the largest integer that is

less than or equal to x.

ex 6

ex 7

ex 8

ex 9

[[ 5 ]] =

[[ 5.999 ]] =

[[ 3 ]] =

[[ -2.6 ]] =

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Theorem 1:

lim f ( x) = L if and only if lim? f ( x) = L = lim f ( x)

x ¡úa

x¡úa +

x¡úa

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

x

ex#10 Prove that the lim does not exist.

x¡ú0 x

Math131 Calculus I

Notes 2.3

page 4

ex#11 What is lim [[ x ]] ?

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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Theorem 2:

If f ( x) ¡Ü g ( x) when x is near a (except possibly at a) and the limits of f and g both

exist as x approaches a then lim f ( x ) ¡Ü lim g ( x ) .

x¡úa

x¡ú a

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1

ex12 Find lim x 2 sin . To find this limit, let¡¯s start by graphing it. Use your graphing calculator.

x ¡ú0

x

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The Squeeze Theorem:

If f ( x) ¡Ü g ( x) ¡Ü h( x) when x is near a (except possibly at a) and

lim f ( x) = lim h( x) = L then lim g ( x) = L

x¡úa

x ¡úa

x¡úa

Math131 Calculus I

Limits at Infinity & Horizontal Asymptotes

Notes 2.6

Definitions of Limits at Large Numbers

numbers

Let f be a function defined on some interval (a, ¡Þ).

Then lim f ( x ) = L means that the values of f(x) can

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numbers

Large

Large

NEGATIVE

POSITIVE

Definition in Words

Horizontal Asymptote

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be made arbitrarily close to L by taking x sufficiently

large in a positive direction.

corresponding number N such that if x > N then

Let f be a function defined on some interval

(-¡Þ,a). ¡Þ). Then lim f ( x) = L means that the

Let f be a function defined on some interval

(-¡Þ,a). Then lim f ( x) = L if for every ¦Å > 0 there

values of f(x) can be made arbitrarily close to L by

taking x sufficiently large in a negative direction.

is a corresponding number N such that if x < N then

x ¡ú ?¡Þ

Definition

Vertical Asymptote

Precise Mathematical Definition

Let f be a function defined on some interval (a, ¡Þ).

Then lim f ( x ) = L if for every ¦Å > 0 there is a

f ( x) ? L < ¦Å

x ¡ú ?¡Þ

f ( x) ? L < ¦Å

What this can look like¡­

The line y = L is a

horizontal asymptote

of the curve y = f(x) if

either is true:

1. lim f ( x) = L

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or

2. lim f ( x) = L

x ¡ú ?¡Þ

The line x = a is a

vertical asymptote

of the curve y = f(x)

if at least one of the

following is true:

1. lim f ( x) = ¡Þ

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2. lim? f ( x) = ¡Þ

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3. lim+ f ( x) = ¡Þ

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4. lim f ( x) = ?¡Þ

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5. lim? f ( x) = ?¡Þ

x¡ú a

6. lim+ f ( x) = ?¡Þ

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Theorem

1

=0

x¡ú¡Þ x r

?

If r > 0 is a rational number then lim

?

If r > 0 is a rational number such that x r is defined for all x then lim

1

=0

x ¡ú ?¡Þ x r

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