Solutions to Homework 4
February 9, 2009
20.10
Prove that
a:
lim
x
→∞
f
(
x
) =
∞
,
b:
lim
x
→
0

f
(
x
) =
−∞
,
c:
lim
x
→
0
+
f
(
x
) =
∞
,
d:
lim
x
→
0
f
(
x
) Does not exists, and
e:
lim
x
→∞
f
(
x
) =
−∞
for
f
(
x
) =
1

x
2
x
.
You can do this by
ǫ

δ
or through the sequence definition.
Proof:
a:
Let
x
n
be a sequence in (
−∞
,
0) such that
x
n
→ −∞
, for any 1
−
M <
0 there is an
integer
N >
0 so that for all
n > N
implies
x
n
<
−
M
+ 1
⇒
M <
−
1
−
x
n
≤
1
/x
n
−
x
n
=
f
(
x
n
)
Since this is true for any sequens in the domain that diverges to
−
infty
lim
x
→∞
f
(
x
) =
∞
b:
Let
x
n
be a sequence in (
−
1
,
0) such that
x
n
→
0, for any
M >
0 there is an integer
N >
0 so
that for all
n > N
implies
0
> x
n
>
−
1
/M
⇒ −
M >
1
/x
n
⇒ −
M
+ 1
>
1
/x
n
+ 1
≤
1
/x
n
−
x
n
=
f
(
x
n
)
Since this is true for any sequens in the domain that converge to 0 lim
x
→∞
f
(
x
) =
−∞
c:
Let
x
n
be a sequence in (0
,
1) such that
x
n
→
0, for any
M >
0 there is an integer
N >
0 so that
for all
n > N
implies
0
< x
n
<
1
/M
⇒
M <
1
/x
n
⇒
M
−
1
>
1
/x
n
−
1
≤
1
/x
n
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 Spring '09
 Logic, The Domain, Sydney, Xn, Dominated convergence theorem

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