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
x→ a
Limit Law in symbols
x→ a
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
x→a
x→a
x→a
x→a
x→a
x→a
x→a
x→a
x→a
lim
x→a
x→a
x→a
f ( x)
f ( x ) lim
= x→a
g ( x ) lim g ( x)
(if lim g ( x) ≠ 0)
x→a
The limit of a quotient is equal to
the quotient of the limits.
x→a
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→a
x→a
x→a
x→a
x→a
x→a
x→a
x→a
x→ a
Direct Substitution Property:
If f is a polynomial or rational function and a is in the domain of f,
then lim f ( x ) =
x→a
“Simpler Function Property”:
If f ( x) = g ( x) when x ≠ a then lim f ( x ) = lim g ( x ) , as long as the
x→a
limit exists.
x→a
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 →3
ex#2
x →3
x →3
2
Evaluate lim 22 x + 1 , if it exists, by using the Limit Laws.
x→2
ex#3
page 2
Evaluate:
x + 6x − 4
lim 2 x 2 + 3 x − 5
x →1
ex#4
Evaluate:
1 − (1 − x) 2
lim
x→0
x
ex#5
Evaluate:
lim
h →0
h+4 −2
h
x →3
Math131 Calculus I
Notes 2.3
page 3
Two Interesting Functions
1.
Absolute Value Function
Definition: x =  x if x ≥ 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 →a
x→a +
x→a
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
x
ex#10 Prove that the lim does not exist.
x→0 x
Math131 Calculus I
Notes 2.3
page 4
ex#11 What is lim [[ x ]] ?
x→3
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Theorem 2:
If f ( x) ≤ 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 ) ≤ lim g ( x ) .
x→a
x→ a
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1
ex12 Find lim x 2 sin . To find this limit, let’s start by graphing it. Use your graphing calculator.
x →0
x
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The Squeeze Theorem:
If f ( x) ≤ g ( x) ≤ 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→a
x →a
x→a
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, ∞).
Then lim f ( x ) = L means that the values of f(x) can
x→∞
numbers
Large
Large
NEGATIVE
POSITIVE
Definition in Words
Horizontal Asymptote
x→∞
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
(-∞,a). ∞). Then lim f ( x) = L means that the
Let f be a function defined on some interval
(-∞,a). Then lim f ( x) = L if for every ε > 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 → −∞
Definition
Vertical Asymptote
Precise Mathematical Definition
Let f be a function defined on some interval (a, ∞).
Then lim f ( x ) = L if for every ε > 0 there is a
f ( x) − L < ε
x → −∞
f ( x) − L < ε
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
x→∞
or
2. lim f ( x) = L
x → −∞
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) = ∞
x→a
2. lim− f ( x) = ∞
x→ a
3. lim+ f ( x) = ∞
x→ a
4. lim f ( x) = −∞
x→a
5. lim− f ( x) = −∞
x→ a
6. lim+ f ( x) = −∞
x→ a
Theorem
1
=0
x→∞ 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 → −∞ x r
Math131 Calculus I
Notes 2.6
3
x→∞ x 5
ex#1
Find the limit: lim
ex#2
Find the limit: lim
ex#3
Find the limit: lim
ex#4
Find the limit: lim cos x
x3 + 2x
x→ ∞ 5 x 3 − x 2 + 4
x→ ∞
x →∞
9 x 2 + x − 3x
page 2
Math131 Calculus I
ex#5
Notes 2.6
Find the vertical and horizontal asymptotes of the graph of the function: f ( x) =
page 3
2x 2 + 1
3x − 5
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