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Limits
Limit # 1.
lim
n
→∞
n
sin
1
n
= lim
x
→∞
x
sin
1
x
= lim
x
→∞
sin
1
x
1
x
= lim
θ
→
0
sin
θ
θ
= 1 by Red #2, p 190.
Deﬁnition 1.
A
sequence
is a function whose domain is the set of all positive
integers.
Deﬁnition 2.
Let
{
a
n
}
be a sequence of real numbers and let
L
be a real number.
Then
(a)
lim
n
→∞
a
n
=
L
iF the following condition holds:
±or each open interval
U
centered at
L,
a
n
is in
U
from some index
N
onward.
(b)
lim
n
→∞
a
n
=
∞
iF the following holds:
±or each interval
U
= (
M,
∞
],
a
n
is in
U
from some index onward.
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This note was uploaded on 05/17/2011 for the course MAC 2312 taught by Professor Bonner during the Spring '08 term at University of Florida.
 Spring '08
 Bonner
 Calculus, Real Numbers, Integers, Limits

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