<s>
Limits	O
are	O
essential	O
to	O
calculus	O
and	O
mathematical	O
analysis	O
,	O
and	O
are	O
used	O
to	O
define	O
continuity	O
,	O
derivatives	B-Algorithm
,	O
and	O
integrals	O
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
is	O
further	O
generalized	O
to	O
the	O
concept	O
of	O
a	O
limit	O
of	O
a	O
topological	O
net	O
,	O
and	O
is	O
closely	O
related	O
to	O
limit	O
and	O
direct	O
limit	O
in	O
category	O
theory	O
.	O
</s>
<s>
The	O
modern	O
definition	O
of	O
a	O
limit	O
goes	O
back	O
to	O
Bernard	O
Bolzano	O
who	O
,	O
in	O
1817	O
,	O
developed	O
the	O
basics	O
of	O
the	O
epsilon-delta	B-Algorithm
technique	O
to	O
define	O
continuous	O
functions	O
.	O
</s>
<s>
Augustin-Louis	O
Cauchy	O
in	O
1821	O
,	O
followed	O
by	O
Karl	O
Weierstrass	O
,	O
formalized	O
the	O
definition	O
of	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
which	O
became	O
known	O
as	O
the	O
( ε	O
,	O
δ	O
)	O
-definition	O
of	O
limit	O
.	O
</s>
<s>
If	O
it	O
does	O
,	O
then	O
it	O
is	O
called	O
convergent	B-Algorithm
,	O
and	O
if	O
it	O
does	O
not	O
,	O
then	O
it	O
is	O
divergent	O
.	O
</s>
<s>
One	O
can	O
show	O
that	O
a	O
convergent	B-Algorithm
sequence	I-Algorithm
has	O
only	O
one	O
limit	O
.	O
</s>
<s>
The	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
and	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
are	O
closely	O
related	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
is	O
the	O
domain	O
of	O
a	O
function	O
and	O
if	O
the	O
limit	O
as	O
approaches	O
infinity	O
of	O
is	O
for	O
every	O
arbitrary	O
sequence	O
of	O
points	O
in	O
which	O
converges	B-Algorithm
to	O
,	O
then	O
the	O
limit	O
of	O
the	O
function	O
as	O
approaches	O
is	O
.	O
</s>
<s>
The	O
sequence	O
of	O
points	O
converges	B-Algorithm
to	O
if	O
the	O
limit	O
exists	O
and	O
.	O
</s>
<s>
If	O
is	O
a	O
topological	O
space	O
with	O
topology	B-Architecture
,	O
and	O
is	O
a	O
sequence	O
in	O
,	O
then	O
the	O
limit	O
(	O
when	O
it	O
exists	O
)	O
of	O
the	O
sequence	O
is	O
a	O
point	O
such	O
that	O
,	O
given	O
a	O
(	O
open	O
)	O
neighborhood	O
of	O
,	O
there	O
exists	O
an	O
such	O
that	O
for	O
every	O
,	O
</s>
<s>
The	O
field	O
of	O
functional	B-Application
analysis	I-Application
partly	O
seeks	O
to	O
identify	O
useful	O
notions	O
of	O
convergence	B-Algorithm
on	O
function	O
spaces	O
.	O
</s>
<s>
Then	O
the	O
sequence	O
is	O
said	O
to	O
converge	B-Algorithm
pointwise	O
to	O
.	O
</s>
<s>
Another	O
notion	O
of	O
convergence	B-Algorithm
is	O
uniform	B-Algorithm
convergence	I-Algorithm
.	O
</s>
<s>
Then	O
the	O
sequence	O
is	O
said	O
to	O
uniformly	O
converge	B-Algorithm
or	O
have	O
a	O
uniform	B-Algorithm
limit	I-Algorithm
of	O
if	O
with	O
respect	O
to	O
this	O
distance	O
.	O
</s>
<s>
The	O
uniform	B-Algorithm
limit	I-Algorithm
has	O
"	O
nicer	O
"	O
properties	O
than	O
the	O
pointwise	O
limit	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
uniform	B-Algorithm
limit	I-Algorithm
of	O
a	O
sequence	O
of	O
continuous	O
functions	O
is	O
continuous	O
.	O
</s>
<s>
Many	O
different	O
notions	O
of	O
convergence	B-Algorithm
can	O
be	O
defined	O
on	O
function	O
spaces	O
.	O
</s>
<s>
Prominent	O
examples	O
of	O
function	O
spaces	O
with	O
some	O
notion	O
of	O
convergence	B-Algorithm
are	O
Lp	O
spaces	O
and	O
Sobolev	O
space	O
.	O
</s>
<s>
In	O
non-standard	O
analysis	O
(	O
which	O
involves	O
a	O
hyperreal	O
enlargement	O
of	O
the	O
number	O
system	O
)	O
,	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
can	O
be	O
expressed	O
as	O
the	O
standard	O
part	O
of	O
the	O
value	O
of	O
the	O
natural	O
extension	O
of	O
the	O
sequence	O
at	O
an	O
infinite	O
hypernatural	O
index	O
n	O
=	O
H	O
.	O
</s>
<s>
The	O
limit	O
set	O
is	O
the	O
set	O
of	O
points	O
such	O
that	O
if	O
there	O
is	O
a	O
convergent	B-Algorithm
subsequence	O
with	O
,	O
then	O
belongs	O
to	O
the	O
limit	O
set	O
.	O
</s>
<s>
A	O
particular	O
expression	O
of	O
interest	O
which	O
is	O
formalized	O
as	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
is	O
sums	O
of	O
infinite	O
series	O
.	O
</s>
<s>
However	O
,	O
while	O
for	O
sequences	O
there	O
is	O
essentially	O
a	O
unique	O
notion	O
of	O
convergence	B-Algorithm
,	O
for	O
series	O
there	O
are	O
different	O
notions	O
of	O
convergence	B-Algorithm
.	O
</s>
<s>
This	O
is	O
due	O
to	O
the	O
fact	O
that	O
the	O
expression	O
does	O
not	O
discriminate	O
between	O
different	O
orderings	O
of	O
the	O
sequence	O
,	O
while	O
the	O
convergence	B-Algorithm
properties	O
of	O
the	O
sequence	O
of	O
partial	O
sums	O
can	O
depend	O
on	O
the	O
ordering	O
of	O
the	O
sequence	O
.	O
</s>
<s>
A	O
series	O
which	O
converges	B-Algorithm
for	O
all	O
orderings	O
is	O
called	O
unconditionally	O
convergent	B-Algorithm
.	O
</s>
<s>
It	O
can	O
be	O
proven	O
to	O
be	O
equivalent	O
to	O
absolute	O
convergence	B-Algorithm
.	O
</s>
<s>
A	O
series	O
is	O
absolutely	O
convergent	B-Algorithm
if	O
is	O
well	O
defined	O
.	O
</s>
<s>
Otherwise	O
,	O
the	O
series	O
is	O
conditionally	O
convergent	B-Algorithm
.	O
</s>
<s>
A	O
surprising	O
result	O
for	O
conditionally	O
convergent	B-Algorithm
series	I-Algorithm
is	O
the	O
Riemann	O
series	O
theorem	O
:	O
depending	O
on	O
the	O
ordering	O
,	O
the	O
partial	O
sums	O
can	O
be	O
made	O
to	O
converge	B-Algorithm
to	O
any	O
real	O
number	O
,	O
as	O
well	O
as	O
.	O
</s>
<s>
Often	O
is	O
thought	O
of	O
as	O
a	O
complex	O
number	O
,	O
and	O
a	O
suitable	O
notion	O
of	O
convergence	B-Algorithm
of	O
complex	O
sequences	O
is	O
needed	O
.	O
</s>
<s>
The	O
set	O
of	O
values	O
of	O
for	O
which	O
the	O
series	O
sum	O
converges	B-Algorithm
is	O
a	O
circle	O
,	O
with	O
its	O
radius	O
known	O
as	O
the	O
radius	O
of	O
convergence	B-Algorithm
.	O
</s>
<s>
then	O
is	O
not	O
defined	O
(	O
see	O
Indeterminate	B-Algorithm
form	I-Algorithm
)	O
,	O
yet	O
as	O
moves	O
arbitrarily	O
close	O
to	O
1	O
,	O
correspondingly	O
approaches	O
2	O
:	O
</s>
<s>
By	O
the	O
convergence	B-Algorithm
of	O
to	O
,	O
there	O
exists	O
an	O
such	O
that	O
for	O
,	O
we	O
have	O
.	O
</s>
<s>
In	O
topology	B-Architecture
,	O
limits	O
are	O
used	O
to	O
define	O
limit	O
points	O
of	O
a	O
subset	O
of	O
a	O
topological	O
space	O
,	O
which	O
in	O
turn	O
give	O
a	O
useful	O
characterization	O
of	O
closed	O
sets	O
.	O
</s>
<s>
The	O
derivative	B-Algorithm
is	O
defined	O
formally	O
as	O
a	O
limit	O
.	O
</s>
<s>
In	O
the	O
scope	O
of	O
real	O
analysis	O
,	O
the	O
derivative	B-Algorithm
is	O
first	O
defined	O
for	O
real	O
functions	O
defined	O
on	O
a	O
subset	O
.	O
</s>
<s>
The	O
derivative	B-Algorithm
at	O
is	O
defined	O
as	O
follows	O
.	O
</s>
<s>
as	O
exists	O
,	O
then	O
the	O
derivative	B-Algorithm
at	O
is	O
this	O
limit	O
.	O
</s>
<s>
If	O
the	O
derivative	B-Algorithm
exists	O
,	O
it	O
is	O
commonly	O
denoted	O
by	O
.	O
</s>
<s>
A	O
property	O
of	O
convergent	B-Algorithm
sequences	I-Algorithm
of	O
real	O
numbers	O
is	O
that	O
they	O
are	O
Cauchy	O
sequences	O
.	O
</s>
<s>
Cauchy	O
sequences	O
are	O
closely	O
related	O
to	O
convergent	B-Algorithm
sequences	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
for	O
sequences	O
of	O
real	O
numbers	O
they	O
are	O
equivalent	O
:	O
any	O
Cauchy	O
sequence	O
is	O
convergent	B-Algorithm
.	O
</s>
<s>
In	O
general	O
metric	O
spaces	O
,	O
it	O
continues	O
to	O
hold	O
that	O
convergent	B-Algorithm
sequences	I-Algorithm
are	O
also	O
Cauchy	O
.	O
</s>
<s>
But	O
the	O
converse	O
is	O
not	O
true	O
:	O
not	O
every	O
Cauchy	O
sequence	O
is	O
convergent	B-Algorithm
in	O
a	O
general	O
metric	O
space	O
.	O
</s>
<s>
The	O
sequence	O
of	O
decimal	O
approximations	O
to	O
,	O
truncated	O
at	O
the	O
th	O
decimal	O
place	O
is	O
a	O
Cauchy	O
sequence	O
,	O
but	O
does	O
not	O
converge	B-Algorithm
in	O
.	O
</s>
<s>
A	O
metric	O
space	O
in	O
which	O
every	O
Cauchy	O
sequence	O
is	O
also	O
convergent	B-Algorithm
,	O
that	O
is	O
,	O
Cauchy	O
sequences	O
are	O
equivalent	O
to	O
convergent	B-Algorithm
sequences	I-Algorithm
,	O
is	O
known	O
as	O
a	O
complete	O
metric	O
space	O
.	O
</s>
<s>
One	O
reason	O
Cauchy	O
sequences	O
can	O
be	O
"	O
easier	O
to	O
work	O
with	O
"	O
than	O
convergent	B-Algorithm
sequences	I-Algorithm
is	O
that	O
they	O
are	O
a	O
property	O
of	O
the	O
sequence	O
alone	O
,	O
while	O
convergent	B-Algorithm
sequences	I-Algorithm
require	O
not	O
just	O
the	O
sequence	O
but	O
also	O
the	O
limit	O
of	O
the	O
sequence	O
.	O
</s>
<s>
Beyond	O
whether	O
or	O
not	O
a	O
sequence	O
converges	B-Algorithm
to	O
a	O
limit	O
,	O
it	O
is	O
possible	O
to	O
describe	O
how	O
fast	O
a	O
sequence	O
converges	B-Algorithm
to	O
a	O
limit	O
.	O
</s>
<s>
One	O
way	O
to	O
quantify	O
this	O
is	O
using	O
the	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
of	O
a	O
sequence	O
.	O
</s>
<s>
A	O
formal	O
definition	O
of	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
can	O
be	O
stated	O
as	O
follows	O
.	O
</s>
<s>
Suppose	O
is	O
a	O
sequence	O
of	O
real	O
numbers	O
which	O
is	O
convergent	B-Algorithm
with	O
limit	O
.	O
</s>
<s>
then	O
is	O
said	O
to	O
converge	B-Algorithm
to	O
with	O
order	B-Architecture
of	I-Architecture
convergence	I-Architecture
.	O
</s>
<s>
Order	B-Architecture
of	I-Architecture
convergence	I-Architecture
is	O
used	O
for	O
example	O
the	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
in	O
error	O
analysis	O
.	O
</s>
<s>
There	O
exist	O
limit	O
expressions	O
whose	O
modulus	O
of	O
convergence	B-Algorithm
is	O
undecidable	O
.	O
</s>
<s>
These	O
are	O
known	O
as	O
convergence	B-Algorithm
tests	O
.	O
</s>
<s>
Examples	O
include	O
the	O
ratio	O
test	O
and	O
the	O
squeeze	B-Algorithm
theorem	I-Algorithm
.	O
</s>
