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Math 310 
hw
9 solutions
20.14, 20.20.a, 28.2;
Friday, 20 Nov 2009
20.14 Prove that
lim
x
→
0
+
1
x
= +
∞
and
lim
x
→
0

1
x
=
∞
.
Let (
x
n
)
n
be a sequence in (0
,
∞
) such that
x
n
→
0. We prove that
1
x
n
→
+
∞
. Let
M >
0, then since
x
n
→
0,
there is
N
such that for all
n
∈
N
,
n > N
implies
x
n
=

x
n

0

<
1
M
.
Hence, for
n
∈
N
,
n > N
implies
1
x
n
> M
. Thus,
1
x
n
→
+
∞
. Therefore, by Deﬁnitions 20.1 and 20.3(b), we have
lim
x
→
0
+
1
x
= +
∞
.
Note that one may also use Theorem 9.10.
The second assertion is proved similarly. Let (
x
n
)
n
be a sequence in (
∞
,
0) such that
x
n
→
0. We prove that
1
x
n
→ ∞
. Let
M <
0, then since
x
n
→
0, there is
N
such that for all
n
∈
N
,
n > N
implies

x
n
=

x
n

0

<

1
M
.
Hence, for
n
∈
N
,
n > N
implies
1
x
n
< M
. Thus,
1
x
n
→ ∞
. Therefore, by Deﬁnitions 20.1 and 20.3(b), we have
lim
x
→
0

1
x
=
∞
.
±
20.20.a Let
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