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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lim
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f ( x)
f ( x ) lim
= x¡úa
g ( x ) lim g ( x)
(if lim g ( x) ¡Ù 0)
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The limit of a quotient is equal to
the quotient of the limits.
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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
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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)
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ex#2
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2
Evaluate lim 22 x + 1 , if it exists, by using the Limit Laws.
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ex#3
page 2
Evaluate:
x + 6x ? 4
lim 2 x 2 + 3 x ? 5
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ex#4
Evaluate:
1 ? (1 ? x) 2
lim
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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 ]] =
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Theorem 1:
lim f ( x) = L if and only if lim? f ( x) = L = lim f ( x)
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x
ex#10 Prove that the lim does not exist.
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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 ) .
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1
ex12 Find lim x 2 sin . To find this limit, let¡¯s start by graphing it. Use your graphing calculator.
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x
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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
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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
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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
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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
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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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