<s>
An	O
iterated	B-Algorithm
limit	I-Algorithm
is	O
only	O
defined	O
for	O
an	O
expression	O
whose	O
value	O
depends	O
on	O
at	O
least	O
two	O
variables	O
.	O
</s>
<s>
To	O
evaluate	O
such	O
a	O
limit	B-Algorithm
,	O
one	O
takes	O
the	O
limiting	O
process	O
as	O
one	O
of	O
the	O
two	O
variables	O
approaches	O
some	O
number	O
,	O
getting	O
an	O
expression	O
whose	O
value	O
depends	O
only	O
on	O
the	O
other	O
variable	O
,	O
and	O
then	O
one	O
takes	O
the	O
limit	B-Algorithm
as	O
the	O
other	O
variable	O
approaches	O
some	O
number	O
.	O
</s>
<s>
This	O
section	O
introduces	O
definitions	O
of	O
iterated	B-Algorithm
limits	I-Algorithm
in	O
two	O
variables	O
.	O
</s>
<s>
The	O
limit(s )	O
for	O
x	O
and/or	O
y	O
can	O
also	O
be	O
taken	O
at	O
infinity	O
,	O
i.e.	O
,	O
</s>
<s>
The	O
limit	B-Algorithm
in	O
x	O
can	O
also	O
be	O
taken	O
at	O
infinity	O
,	O
i.e.	O
,	O
</s>
<s>
Note	O
that	O
the	O
limit	B-Algorithm
in	O
n	O
is	O
taken	O
discretely	O
,	O
while	O
the	O
limit	B-Algorithm
in	O
x	O
is	O
taken	O
continuously	O
.	O
</s>
<s>
The	O
following	O
theorem	O
states	O
the	O
relationship	O
between	O
double	O
limit	B-Algorithm
and	O
iterated	B-Algorithm
limits	I-Algorithm
.	O
</s>
<s>
For	O
this	O
limit	B-Algorithm
to	O
exist	O
,	O
f(x, y )	O
can	O
be	O
made	O
as	O
close	O
to	O
L	O
as	O
desired	O
along	O
every	O
possible	O
path	O
approaching	O
the	O
point	O
(	O
a	O
,	O
b	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
value	O
of	O
f	O
at	O
the	O
point	O
(	O
a	O
,	O
b	O
)	O
,	O
even	O
if	O
it	O
is	O
defined	O
,	O
does	O
not	O
affect	O
the	O
limit	B-Algorithm
.	O
</s>
<s>
For	O
this	O
limit	B-Algorithm
to	O
exist	O
,	O
f(x, y )	O
can	O
be	O
made	O
as	O
close	O
to	O
L	O
as	O
desired	O
along	O
every	O
possible	O
path	O
approaching	O
the	O
point	O
(	O
a	O
,	O
b	O
)	O
,	O
except	O
the	O
lines	O
x	O
=	O
a	O
and	O
y	O
=	O
b	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
value	O
of	O
f	O
along	O
the	O
lines	O
x	O
=	O
a	O
and	O
y	O
=	O
b	O
does	O
not	O
affect	O
the	O
limit	B-Algorithm
.	O
</s>
<s>
This	O
is	O
different	O
from	O
the	O
ordinary	O
limit	B-Algorithm
where	O
only	O
the	O
point	O
(	O
a	O
,	O
b	O
)	O
is	O
excluded	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
ordinary	O
limit	B-Algorithm
is	O
a	O
stronger	O
notion	O
than	O
double	O
limit	B-Algorithm
:	O
</s>
<s>
Both	O
of	O
these	O
limits	O
do	O
not	O
involve	O
first	O
taking	O
one	O
limit	B-Algorithm
and	O
then	O
another	O
.	O
</s>
<s>
This	O
contrasts	O
with	O
iterated	B-Algorithm
limits	I-Algorithm
where	O
the	O
limiting	O
process	O
is	O
taken	O
in	O
x-direction	O
first	O
,	O
and	O
then	O
in	O
y-direction	O
(	O
or	O
in	O
reversed	O
order	O
)	O
.	O
</s>
<s>
The	O
following	O
theorem	O
states	O
the	O
relationship	O
between	O
double	O
limit	B-Algorithm
and	O
iterated	B-Algorithm
limits	I-Algorithm
:	O
</s>
<s>
The	O
following	O
theorem	O
states	O
the	O
relationship	O
between	O
double	O
limit	B-Algorithm
at	O
infinity	O
and	O
iterated	B-Algorithm
limits	I-Algorithm
at	O
infinity	O
:	O
</s>
<s>
The	O
converses	O
of	O
Theorems	O
1	O
,	O
3	O
and	O
4	O
do	O
not	O
hold	O
,	O
i.e.	O
,	O
the	O
existence	O
of	O
iterated	B-Algorithm
limits	I-Algorithm
,	O
even	O
if	O
they	O
are	O
equal	O
,	O
does	O
not	O
imply	O
the	O
existence	O
of	O
the	O
double	O
limit	B-Algorithm
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
the	O
double	O
limit	B-Algorithm
does	O
not	O
exist	O
.	O
</s>
<s>
The	O
essence	O
of	O
the	O
interchangeability	O
depends	O
on	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
If	O
uniformly	O
(	O
in	O
m	O
)	O
,	O
and	O
for	O
each	O
large	O
n	O
,	O
then	O
both	O
and	O
exists	O
and	O
are	O
equal	O
to	O
the	O
double	O
limit	B-Algorithm
,	O
i.e.	O
,	O
</s>
<s>
By	O
the	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
for	O
any	O
there	O
exist	O
such	O
that	O
for	O
all	O
,	O
implies	O
.	O
</s>
<s>
As	O
,	O
we	O
have	O
,	O
which	O
means	O
that	O
is	O
a	O
Cauchy	O
sequence	O
which	O
converges	O
to	O
a	O
limit	B-Algorithm
.	O
</s>
<s>
Also	O
,	O
by	O
taking	O
,	O
we	O
see	O
that	O
this	O
limit	B-Algorithm
also	O
equals	O
.	O
</s>
<s>
If	O
converges	B-Algorithm
uniformly	I-Algorithm
(	O
in	O
m	O
)	O
,	O
and	O
converges	O
for	O
each	O
large	O
n	O
,	O
then	O
.	O
</s>
<s>
If	O
uniformly	O
(	O
in	O
y	O
)	O
on	O
,	O
and	O
for	O
each	O
x	O
near	O
a	O
,	O
then	O
both	O
and	O
exists	O
and	O
are	O
equal	O
to	O
the	O
double	O
limit	B-Algorithm
,	O
i.e.	O
,	O
</s>
<s>
By	O
the	O
existence	O
uniform	B-Algorithm
limit	I-Algorithm
,	O
for	O
any	O
there	O
exist	O
such	O
that	O
for	O
all	O
,	O
and	O
implies	O
.	O
</s>
<s>
By	O
the	O
existence	O
of	O
pointwise	O
limit	B-Algorithm
,	O
for	O
any	O
and	O
near	O
,	O
there	O
exist	O
such	O
that	O
implies	O
.	O
</s>
<s>
Also	O
,	O
by	O
taking	O
,	O
we	O
see	O
that	O
this	O
limit	B-Algorithm
also	O
equals	O
.	O
</s>
<s>
By	O
the	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
for	O
any	O
there	O
exist	O
such	O
that	O
for	O
all	O
,	O
implies	O
.	O
</s>
<s>
As	O
,	O
we	O
have	O
,	O
which	O
means	O
that	O
is	O
a	O
Cauchy	O
sequence	O
which	O
converges	O
to	O
a	O
limit	B-Algorithm
.	O
</s>
<s>
By	O
the	O
existence	O
of	O
pointwise	O
limit	B-Algorithm
,	O
for	O
any	O
and	O
,	O
there	O
exist	O
such	O
that	O
implies	O
.	O
</s>
<s>
A	O
corollary	O
is	O
the	O
continuity	O
theorem	O
for	O
uniform	B-Algorithm
convergence	I-Algorithm
as	O
follows	O
:	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
uniform	B-Algorithm
limit	I-Algorithm
of	O
continuous	O
functions	O
is	O
continuous	O
.	O
</s>
<s>
Another	O
corollary	O
is	O
about	O
the	O
interchangeability	O
of	O
limit	B-Algorithm
and	O
infinite	O
sum	O
.	O
</s>
<s>
If	O
converges	B-Algorithm
uniformly	I-Algorithm
(	O
in	O
x	O
)	O
on	O
,	O
and	O
exists	O
for	O
each	O
large	O
n	O
,	O
then	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
double	O
limit	B-Algorithm
does	O
not	O
exist	O
,	O
and	O
thus	O
this	O
problem	O
is	O
not	O
well-defined	O
.	O
</s>
<s>
By	O
the	O
integration	O
theorem	O
for	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
once	O
we	O
have	O
converges	B-Algorithm
uniformly	I-Algorithm
on	O
,	O
the	O
limit	B-Algorithm
in	O
n	O
and	O
an	O
integration	O
over	O
a	O
bounded	O
interval	O
can	O
be	O
interchanged	O
:	O
</s>
<s>
By	O
Weierstrass	B-Algorithm
M-test	I-Algorithm
,	O
converges	B-Algorithm
uniformly	I-Algorithm
on	O
.	O
</s>
<s>
Then	O
by	O
the	O
integration	O
theorem	O
for	O
uniform	B-Algorithm
convergence	I-Algorithm
,	O
.	O
</s>
<s>
To	O
further	O
interchange	O
the	O
limit	B-Algorithm
with	O
the	O
infinite	O
summation	O
,	O
the	O
Moore-Osgood	O
theorem	O
requires	O
the	O
infinite	O
series	O
to	O
be	O
uniformly	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
Again	O
,	O
by	O
Weierstrass	B-Algorithm
M-test	I-Algorithm
,	O
converges	B-Algorithm
uniformly	I-Algorithm
on	O
.	O
</s>
