Problem Set 2
MTH211
Solutions
1 Questions on understanding the lecture
E±²³´µ¶² · What are the domain, the codomain, and the component functions of the
following maps?
(a)
±
(
²³´³µ
) =
¸
²
+
´
´ 
sin
µ
¹
;
(b)
±
(
²³´
) =
º
1
 ²
2
 ´
2
;
(c)
±
(
¶³·
) =
¶·
3
sin(
¶·
)
ln(
¶·
)
¸
S»¼½¾µ»¿
(a) The component functions are
±
1
(
) =
²
+
´³±
2
(
) =
´ 
sin
µ
and
the codomain is
R
2
. Now we need to ﬁnd the domain, that is, the set where the map
is deﬁned. It’s easy to see that the map is deﬁned for all values of
²
,
´
,
µ
, so the
domain is
R
3
.
(b) The component function is just
±
1
(
) =
º
1
 ²
2
 ´
2
and the codomain is
R
. Now
we need to ﬁnd the domain, that is, the set where the map is deﬁned. Expression
under the square root must be nonnegative, that is, 1
 ²
2
 ´
2
≥
0, or the domain
is
²
2
+
´
2
≤
1. It is geometrically the unit disc centered at (0
³
0) in
R
2
.
(c) The component functions are
±
1
(
¶·
) =
¶·
3
³±
2
(
) = sin(
¶·
)
3
(
) = ln(
¶·
) and
the codomain is
R
3
. Now we need to ﬁnd the domain, that is, the set where the
map is deﬁned. Here,
¶·
3
and sin(
¶·
) are always deﬁned, but for ln(
¶·
) we need
¶· >
0. In other words, we either must have
¶³· >
0 or
¶³· <
0. Thus the domain
of this map is the union of the open I and III quadrants.
:)
1
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View Full DocumentE±²³´µ¶² · Given a map
±
:
R
²
→
R
³
, prove that lim
x
→
a
±
(
x
) =
l
holds if and only if
lim
x
→
a
´±
(
x
)

l
´
= 0 does.
S¸¹º»µ¸¼
We ﬁrst prove the “only if" part. For this, we assume lim
x
→
a
±
(
x
) =
l
holds and
we want to show this implies that lim
x
→
a
´±
(
x
)

l
´
= 0. Now it follows from the deﬁnition
that lim
x
→
a
±
(
x
) =
l
means that for any given
µ >
0, there exists a
δ >
0 such that for
any 0
< ´
x

a
´ < δ
, we have
´±
(
x
)

l
´ < µ
. Note that we have
´±
(
x
)

l
´ ≥
0 so that
´±
(
x
)

l
´
=
´±
(
x
)

l
´ 
0

. Therefore, we see that for any
µ >
0, there exists a
δ >
0
such that for any 0
< ´
x

a
´ < δ
, we have
´±
(
x
)

l
´
0

=
´±
(
x
)

l
´ < µ
, which implies
that lim
x
→
a
´±
(
x
)

l
´
= 0 by deﬁnition.
We now prove the “if" part. For this, we assume lim
x
→
a
´±
(
x
)

l
´
= 0 holds and we want to
show this implies that lim
x
→
a
±
(
x
) =
l
. Now it follows from the deﬁnition that lim
x
→
a
´±
(
x
)

l
´
= 0
means that for any given
µ >
0, there exists a
δ >
0 such that for any 0
< ´
x

a
´ < δ
, we
have
´±
(
x
)

l
0
 < µ
. Note that we have
´±
(
x
)

l
´ ≥
0 so that
´±
(
x
)

l
0

=
´±
(
x
)

l
´
.
Therefore, we see that for any
µ >
0, there exists a
δ >
0 such that for any 0
< ´
x

a
´ < δ
,
we have
´±
(
x
)

l
´ < µ
, which implies that lim
x
→
a
±
(
x
) =
l
by deﬁnition.
:)
2 Questions on calculation
E±²³´µ¶² ½ Prove that
lim
(
¶·¸
)
→
(0
·
0)
¶
2
¸
2
¶
2
¸
2
+ (
¶  ¸
)
2
is not deﬁned (by ﬁnding limits along dif
ferent straight lines).
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 Spring '13
 Limit, lim, Limit of a function

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