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MAT 1234
Calculus I
Section 2.2
The Limit of a Function
http://myhome.spu.edu/lauw
WebAssign
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Homework 1.5
What do we care?
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How fast “things” are going
• The velocity of a particle
• The “speed” of formation of chemicals
• The rate of change of charges in a capacitor
Recall
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Limit of the following form is important
f ( a  h)  f ( a )
lim
h 0
h
Recall
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f ( a  h)  f ( a )
lim
h 0
h
Recall

f ( a  h)  f ( a )
lim
h 0
h
Recall
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Limit of the following form is important
f ( a  h)  f ( a )
lim
h 0
h
1.4: Estimate limits by tables
1.6: Compute limits by algebra
1.5: Formally define limits
Preview
 One-sided
limits
 The existence of limits
 Infinite limits
One-Sided Limits

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We will focus on the conceptual
meaning.
We will not look closely at the formal
definition.
Left-Hand Limit
y


The left-hand limit is 2
when x approaches 3
Notation:
lim f ( x)  2
x 3
2
y=f(x)

3
x
Independent of f(3)
Left-Hand Limit
y

lim f ( x)  2
x 3
2
y=f(x)
3
x
Left-Hand Limit
y

lim f ( x) 
x 3
2
y=f(x)
3
x
Left-Hand Limit
y

lim f ( x) 
x 3
2
y=f(x)
3
x
Left-Hand Limit (Formal)
lim f ( x)  L
xa
Right-Hand Limit
y

4

y=f(x)
The right-hand limit is 4
when x approaches 2
Notation:
lim f ( x)  4
x2

2
x
Independent of f(2)
Right-Hand Limit
y

4
y=f(x)
lim f ( x)  4
x2
2
x
Right-Hand Limit
y

4
y=f(x)
lim f ( x) 
x2
2
x
Right-Hand Limit
y

4
y=f(x)
lim f ( x) 
x2
2
x
Right-Hand Limit (Formal)
lim f ( x)  L
x a
Limit of a Function
lim f ( x)  L
x a
lim f ( x)  L
xa
lim f ( x)  L
xa
Example 1
y
lim f ( x) 
x2
lim f ( x) 
x2
1.5
lim f ( x) 
x2
2
x
Example 2
y
lim f ( x) 
x2
lim f ( x) 
2.5
x2
1.5
lim f ( x) 
x2
2
x
Infinite Limits
y


y=f(x)
The left-hand limit DNE
Notation:
lim f ( x) 
xa
is not a number
a
x
Infinite Limits

Let us look at all possible situations and
their standard representations.
Infinite Limits
y


The left-hand limit DNE
Notation:
lim f ( x) 
xa
a
x
Infinite Limits
y


The right-hand limit DNE
Notation:
lim f ( x) 
xa
a
x
Infinite Limits
y


The right-hand limit DNE
Notation:
lim f ( x) 
xa
a
x
Infinite Limits
y


The limit DNE
Notation:
lim f ( x) 
xa
a
x
Infinite Limits
y


The limit DNE
Notation:
lim f ( x) 
xa
a
x
Infinite Limits
y


The limit DNE
Notation:
lim f ( x) 
x a
lim f ( x) 
x a
a
x
Review: We learned…
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left-hand and right-hand limits
definition of limits
infinite limits
infinity is a concept, not a number
to use infinity to represent geometry
concepts - unboundness
Classwork
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
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
Use pencils!
Work with your partner ONLY.
No discussions between groups
(penalty= last group to get out)
You can go, after Katie or I check both of
your work.