School

Pennsylvania State University **We aren't endorsed by this school

Course

MATH 140B

Subject

Mathematics

Date

Oct 26, 2023

Pages

7

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1
SECTION 3.1 - INTRODUCTION TO LIMITS
I. INTRODUCTION TO LIMITS
Limits describe the behavior of functions
࠵? = ࠵?(࠵?)
near
a specified
࠵?
-value.
Example 1: Consider the function
࠵?(࠵?) =
!
!
"#
!"#
.
a. What happens to
࠵?(࠵?)
when
࠵? = 1
?
b. What happens to
࠵?(࠵?)
when
࠵?
c. What happens to
࠵?(࠵?)
when
࠵?
is
close
to 1, but
less than
1?
is
close
to 1, but
greater than
1?
࠵?
࠵?(࠵?)
0
1
0.9
1.9
0.99
1.99
0.999
1.999
࠵?
࠵?(࠵?)
2
3
1.1
2.1
1.01
2.01
1.001
2.001

2
Important Notes:
1.
If no side indicated, the "limit" is implied to be the (two-sided) limit.
2.
If
lim
!→%
࠵?(࠵?) = ࠵?
and
࠵?
is a
finite number
, we say ________________________.
Otherwise, ______________________________.
3.
Limits describe the behavior of
࠵?(࠵?)
near
࠵? = ࠵?
, not at
࠵? = ࠵?
. In fact, the
function value
࠵?(࠵?)
need not exist
for the limit to exist.
ONE-SIDED AND (TWO-SIDED) LIMITS
1.
If
࠵?(࠵?)
becomes
arbitrarily close to
࠵?
whenever
࠵?
gets
closer to
(but to the
left
of)
࠵?
on the number line, then
࠵?
is the
left-hand limit
. We write
lim
!→%
"
࠵?(࠵?) = ࠵?
2.
If
࠵?(࠵?)
becomes
arbitrarily close to
࠵?
whenever
࠵?
gets
closer to
(but to the
right
of)
࠵?
on the number line, then
࠵?
is the
right-hand limit
. We write
lim
!→%
#
࠵?(࠵?) = ࠵?
3.
If
࠵?(࠵?)
becomes
arbitrarily close
to a single number
࠵?
as
࠵?
approaches
࠵?
from either side
, then the
limit of
࠵?(࠵?)
as
࠵?
approaches
࠵?
is
࠵?
. We write
lim
!→%
࠵?(࠵?) = ࠵?
or
"࠵?(࠵?) → ࠵? as ࠵? → ࠵?"
THEOREM
The limit itself exists if and only if both one-sided limits exist and are
equal. Otherwise, the limit does not exist.
Þ
In other words,
in order for a limit to exist
,

3
Example 2: Find each limit using the graph of
࠵?(࠵?)
shown below.
a.
lim
!→"&
"
࠵?(࠵?) =
e.
lim
!→"#
"
࠵?(࠵?) =
i.
lim
!→#
"
࠵?(࠵?) =
b.
lim
!→"&
#
࠵?(࠵?) =
f.
lim
!→"#
#
࠵?(࠵?) =
j.
lim
!→#
#
࠵?(࠵?) =
c.
lim
!→"&
࠵?(࠵?) =
g.
lim
!→"#
࠵?(࠵?) =
k.
lim
!→#
࠵?(࠵?) =
d.
࠵?(−2) =
h.
࠵?(−1) =
j.
࠵?(1) =
II. INFINITE LIMITS
Example 3: Consider the function
࠵?(࠵?) =
#
!"&
whose graph is shown below.
a. What is
࠵?(2)
?
b. What happens to
࠵?(࠵?)
as
࠵? → 2
"
?
c. What happens to
࠵?(࠵?)
as
࠵? → 2
'
?
d. What is
lim
!→&
࠵?(࠵?) ?

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