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Math 1007, Fall 2006
(These slides replace neither the text book nor the lectures.)
Part II: Limits
11. Limits (1719)
12. Properties of Limits (2021)
(2226)
14. One Sided Limits (2728)
15. Continuity (2931)
16
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View Full Document 11. Limits (2.2)
Consider the function
f
(
x
) =
x
2
.
We wish to determine the value that the function
f
approaches when
x
is close to 2.
Approaching from the left:
x <
2
x
1.75
1.9
1.99
1.999
f
(
x
)
3.0625
3.61
3.9601
3.996001
Approaching from the right:
x >
2
x
2.25
2.1
2.01
2.001
f
(
x
)
5.0625
4.41
4.0401
4.004001
Since
f
approaches 4 as
x
approaches 2, we write
lim
x
→
2
f
(
x
) = lim
x
→
2
x
2
= 4
.
17
De¯nition:
The function
f
has the
limit
L as
x
ap
proaches
a
, written
lim
x
→
a
f
(
x
) =
L
if the value of
f
(
x
) can be made as close to the number
L
as we please by taking
x
su±ciently close to (but not
equal to)
a
.
Example 1:
Consider
g
(
t
) =
4(
t
2
−
4)
t
−
2
.
Note that
g
is
not
de¯ned at
t
= 2.
t
1.9
1.99
1.999
2.001
2.01
2.1
g
(
t
)
15.6
15.96
15.996
16.004
16.04
16.4
Since
g
(
t
) approaches 16 if
t
approaches 2,
lim
t
→
2
g
(
t
) = lim
x
→
2
4(
t
2
−
4)
t
−
2
= 16
.
Hence the limit at
a
may exist even if the function is
not de¯ned at
a
.
18
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View Full Document Example 2:
Consider
g
(
x
) =
b
x
+ 2
x
n
= 1
1
x
= 1
From the graph, we see that with
x
near 1 (but not
equal to 1),
g
(
x
) is near 3 so
lim
x
→
1
g
(
x
) = 3
.
Note that
g
(1) = 1, so the value of the function has no
bearing on the value of the limit.
Example 3:
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This note was uploaded on 03/22/2010 for the course MATH 1007 taught by Professor Unknown during the Fall '07 term at Carleton.
 Fall '07
 Unknown
 Calculus, Continuity, Limits

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