<s>
In	O
mathematics	O
,	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
is	O
a	O
fundamental	O
concept	O
in	O
calculus	O
and	O
analysis	O
concerning	O
the	O
behavior	O
of	O
that	O
function	O
near	O
a	O
particular	O
input	O
.	O
</s>
<s>
The	O
concept	O
of	O
limit	O
also	O
appears	O
in	O
the	O
definition	O
of	O
the	O
derivative	B-Algorithm
:	O
in	O
the	O
calculus	O
of	O
one	O
variable	O
,	O
this	O
is	O
the	O
limiting	O
value	O
of	O
the	O
slope	O
of	O
secant	O
lines	O
to	O
the	O
graph	O
of	O
a	O
function	O
.	O
</s>
<s>
Although	O
implicit	O
in	O
the	O
development	O
of	O
calculus	O
of	O
the	O
17th	O
and	O
18th	O
centuries	O
,	O
the	O
modern	O
idea	O
of	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
goes	O
back	O
to	O
Bolzano	O
who	O
,	O
in	O
1817	O
,	O
introduced	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>
In	O
his	O
1821	O
book	O
Cours	O
d'analyse	O
,	O
Cauchy	O
discussed	O
variable	O
quantities	O
,	O
infinitesimals	O
and	O
limits	O
,	O
and	O
defined	O
continuity	O
of	O
by	O
saying	O
that	O
an	O
infinitesimal	O
change	O
in	O
x	O
necessarily	O
produces	O
an	O
infinitesimal	O
change	O
in	O
y	O
,	O
while	O
claims	O
that	O
he	O
used	O
a	O
rigorous	O
epsilon-delta	B-Algorithm
definition	I-Algorithm
in	O
proofs	O
.	O
</s>
<s>
In	O
1861	O
,	O
Weierstrass	O
first	O
introduced	O
the	O
epsilon-delta	B-Algorithm
definition	I-Algorithm
of	O
limit	O
in	O
the	O
form	O
it	O
is	O
usually	O
written	O
today	O
.	O
</s>
<s>
The	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
f(x )	O
as	O
x	O
approaches	O
p	O
is	O
a	O
number	O
L	O
with	O
the	O
following	O
property	O
:	O
given	O
any	O
target	O
distance	O
from	O
L	O
,	O
there	O
is	O
a	O
distance	O
from	O
p	O
within	O
which	O
the	O
values	O
of	O
f(x )	O
remain	O
within	O
the	O
target	O
distance	O
.	O
</s>
<s>
In	O
fact	O
,	O
this	O
explicit	O
statement	O
is	O
quite	O
close	O
to	O
the	O
formal	O
definition	O
of	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
,	O
with	O
values	O
in	O
a	O
topological	O
space	O
.	O
</s>
<s>
The	O
following	O
definitions	O
,	O
known	O
as	O
( ε	O
,	O
δ	O
)	O
-definitions	O
,	O
are	O
the	O
generally	O
accepted	O
definitions	O
for	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
in	O
various	O
contexts	O
.	O
</s>
<s>
If	O
these	O
limits	O
exist	O
at	O
p	O
and	O
are	O
equal	O
there	O
,	O
then	O
this	O
can	O
be	O
referred	O
to	O
as	O
the	O
limit	O
of	O
f(x )	O
at	O
p	O
.	O
If	O
the	O
one-sided	B-Algorithm
limits	I-Algorithm
exist	O
at	O
p	O
,	O
but	O
are	O
unequal	O
,	O
then	O
there	O
is	O
no	O
limit	O
at	O
p	O
(	O
i.e.	O
,	O
the	O
limit	O
at	O
p	O
does	O
not	O
exist	O
)	O
.	O
</s>
<s>
If	O
either	O
one-sided	B-Algorithm
limit	I-Algorithm
does	O
not	O
exist	O
at	O
p	O
,	O
then	O
the	O
limit	O
at	O
p	O
also	O
does	O
not	O
exist	O
.	O
</s>
<s>
It	O
also	O
extends	O
the	O
notion	O
of	O
one-sided	B-Algorithm
limits	I-Algorithm
to	O
the	O
included	O
endpoints	O
of	O
(	O
half	O
-	O
)	O
closed	O
intervals	O
,	O
so	O
the	O
square	O
root	O
function	O
f(x )	O
=	O
can	O
have	O
limit	O
0	O
as	O
x	O
approaches	O
0	O
from	O
above	O
:	O
</s>
<s>
Limits	O
involving	O
infinity	O
are	O
connected	O
with	O
the	O
concept	O
of	O
asymptotes	B-Language
.	O
</s>
<s>
In	O
certain	O
applications	O
of	O
numerical	B-General_Concept
differentiation	I-General_Concept
and	I-General_Concept
integration	I-General_Concept
,	O
it	O
is	O
,	O
for	O
example	O
,	O
convenient	O
to	O
have	O
signed	B-Algorithm
zeroes	I-Algorithm
.	O
</s>
<s>
If	O
the	O
limit	O
at	O
infinity	O
exists	O
,	O
it	O
represents	O
a	O
horizontal	O
asymptote	B-Language
at	O
y	O
=	O
L	O
.	O
Polynomials	O
do	O
not	O
have	O
horizontal	O
asymptotes	B-Language
;	O
such	O
asymptotes	B-Language
may	O
however	O
occur	O
with	O
rational	O
functions	O
.	O
</s>
<s>
Since	O
cos(θ )	O
is	O
bounded	O
between	O
[ −1	O
,	O
1 ]	O
,	O
by	O
the	O
sandwich	B-Algorithm
theorem	I-Algorithm
,	O
this	O
limit	O
tends	O
to	O
0	O
.	O
</s>
<s>
Hence	O
no	O
matter	O
how	O
y	O
behaves	O
,	O
we	O
may	O
use	O
the	O
sandwich	B-Algorithm
theorem	I-Algorithm
to	O
show	O
that	O
the	O
limit	O
is	O
0	O
.	O
</s>
<s>
Finally	O
,	O
we	O
will	O
discuss	O
the	O
limit	O
in	O
function	B-Algorithm
space	I-Algorithm
,	O
which	O
has	O
infinite	O
dimensions	O
.	O
</s>
<s>
Consider	O
a	O
function	O
f(x, y )	O
in	O
the	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
We	O
want	O
to	O
find	O
out	O
as	O
x	O
approaches	O
p	O
,	O
how	O
f(x, y )	O
will	O
tend	O
to	O
another	O
function	O
g(y )	O
,	O
which	O
is	O
in	O
the	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
"	O
closeness	O
"	O
in	O
this	O
function	B-Algorithm
space	I-Algorithm
may	O
be	O
measured	O
under	O
the	O
uniform	O
metric	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
the	O
domain	O
of	O
f	O
is	O
X{p}	O
(	O
or	O
all	O
of	O
X	O
)	O
,	O
then	O
the	O
limit	O
of	O
f	O
as	O
x	O
→	O
p	O
exists	O
and	O
is	O
equal	O
to	O
L	O
if	O
,	O
for	O
all	O
subsets	O
Ω	O
of	O
X	O
with	O
limit	O
point	O
p	O
,	O
the	O
limit	O
of	O
the	O
restriction	O
of	O
f	O
to	O
Ω	O
exists	O
and	O
is	O
equal	O
to	O
L	O
.	O
Sometimes	O
this	O
criterion	O
is	O
used	O
to	O
establish	O
the	O
non-existence	O
of	O
the	O
two-sided	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
on	O
R	O
by	O
showing	O
that	O
the	O
one-sided	B-Algorithm
limits	I-Algorithm
either	O
fail	O
to	O
exist	O
or	O
do	O
not	O
agree	O
.	O
</s>
<s>
Alternatively	O
,	O
the	O
requirement	O
that	O
Y	O
be	O
a	O
Hausdorff	O
space	O
can	O
be	O
relaxed	O
to	O
the	O
assumption	O
that	O
Y	O
be	O
a	O
general	O
topological	O
space	O
,	O
but	O
then	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
may	O
not	O
be	O
unique	O
.	O
</s>
<s>
In	O
particular	O
,	O
one	O
can	O
no	O
longer	O
talk	O
about	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
at	O
a	O
point	O
,	O
but	O
rather	O
a	O
limit	O
or	O
the	O
set	O
of	O
limits	O
at	O
a	O
point	O
.	O
</s>
<s>
There	O
is	O
another	O
type	O
of	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
,	O
namely	O
the	O
sequential	O
limit	O
.	O
</s>
<s>
For	O
every	O
sequence	O
(	O
xn	O
)	O
in	O
that	O
converges	B-Algorithm
to	O
p	O
,	O
the	O
sequence	O
f(xn )	O
converges	B-Algorithm
to	O
L	O
.	O
</s>
<s>
For	O
functions	O
on	O
the	O
real	O
line	O
,	O
one	O
way	O
to	O
define	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
is	O
in	O
terms	O
of	O
the	O
limit	O
of	O
sequences	O
.	O
</s>
<s>
if	O
,	O
and	O
only	O
if	O
,	O
for	O
all	O
sequences	O
(	O
with	O
not	O
equal	O
to	O
a	O
for	O
all	O
n	O
)	O
converging	O
to	O
the	O
sequence	O
converges	B-Algorithm
to	O
.	O
</s>
<s>
Note	O
that	O
defining	O
what	O
it	O
means	O
for	O
a	O
sequence	O
to	O
converge	O
to	O
requires	O
the	O
epsilon	B-Algorithm
,	I-Algorithm
delta	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Let	O
a	O
be	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
sequence	I-Algorithm
of	O
elements	O
of	O
Dm(f )	O
\{a}	O
.	O
</s>
<s>
if	O
for	O
every	O
sequence	O
∈	O
Dm(f )	O
\{a}	O
(	O
so	O
that	O
for	O
all	O
n	O
,	O
is	O
not	O
equal	O
to	O
a	O
)	O
that	O
converges	B-Algorithm
to	O
a	O
,	O
the	O
sequence	O
converges	B-Algorithm
to	O
.	O
</s>
<s>
In	O
non-standard	O
calculus	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
is	O
defined	O
by	O
:	O
</s>
<s>
The	O
notion	O
of	O
the	O
limit	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
is	O
very	O
closely	O
related	O
to	O
the	O
concept	O
of	O
continuity	O
.	O
</s>
<s>
If	O
f	O
:	O
M	O
→	O
N	O
is	O
a	O
function	O
between	O
metric	O
spaces	O
M	O
and	O
N	O
,	O
then	O
it	O
is	O
equivalent	O
that	O
f	O
transforms	O
every	O
sequence	O
in	O
M	O
which	O
converges	B-Algorithm
towards	O
p	O
into	O
a	O
sequence	O
in	O
N	O
which	O
converges	B-Algorithm
towards	O
f(p )	O
.	O
</s>
<s>
These	O
rules	O
are	O
also	O
valid	O
for	O
one-sided	B-Algorithm
limits	I-Algorithm
,	O
including	O
when	O
p	O
is	O
∞	O
or	O
−∞	O
.	O
</s>
<s>
In	O
other	O
cases	O
the	O
limit	O
on	O
the	O
left	O
may	O
still	O
exist	O
,	O
although	O
the	O
right-hand	O
side	O
,	O
called	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
,	O
does	O
not	O
allow	O
one	O
to	O
determine	O
the	O
result	O
.	O
</s>
<s>
This	O
depends	O
on	O
the	O
functions	O
f	O
and	O
g	O
.	O
These	O
indeterminate	B-Algorithm
forms	I-Algorithm
are	O
:	O
</s>
<s>
See	O
further	O
Hôpital	O
's	O
rule	O
below	O
and	O
Indeterminate	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
This	O
rule	O
uses	O
derivatives	B-Algorithm
to	O
find	O
limits	O
of	O
indeterminate	B-Algorithm
forms	I-Algorithm
or	O
,	O
and	O
only	O
applies	O
to	O
such	O
cases	O
.	O
</s>
<s>
Other	O
indeterminate	B-Algorithm
forms	I-Algorithm
may	O
be	O
manipulated	O
into	O
this	O
form	O
.	O
</s>
