<s>
This	O
is	O
a	O
list	B-Algorithm
of	I-Algorithm
limits	I-Algorithm
for	O
common	O
functions	O
such	O
as	O
elementary	O
functions	O
.	O
</s>
<s>
if	O
and	O
only	O
if	O
This	O
is	O
the	O
( ε	O
,	O
δ	O
)	O
-definition	O
of	O
limit	B-Algorithm
.	O
</s>
<s>
The	O
limit	B-Algorithm
superior	I-Algorithm
and	I-Algorithm
limit	I-Algorithm
inferior	I-Algorithm
of	O
a	O
sequence	O
are	O
defined	O
as	O
and	O
.	O
</s>
<s>
This	O
is	O
the	O
definition	O
of	O
the	O
derivative	B-Algorithm
.	O
</s>
<s>
This	O
is	O
known	O
as	O
the	O
squeeze	B-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
if	O
is	O
a	O
polynomial	O
then	O
,	O
by	O
the	O
continuity	O
of	O
polynomials	O
,	O
This	O
is	O
also	O
true	O
for	O
rational	O
functions	O
,	O
as	O
they	O
are	O
continuous	O
on	O
their	O
domains	B-Algorithm
.	O
</s>
<s>
This	O
limit	B-Algorithm
can	O
be	O
derived	O
from	O
this	O
limit	B-Algorithm
.	O
</s>
<s>
This	O
limit	B-Algorithm
follows	O
from	O
Hôpital	O
's	O
rule	O
.	O
</s>
<s>
In	O
general	O
,	O
any	O
infinite	O
series	O
is	O
the	O
limit	B-Algorithm
of	O
its	O
partial	O
sums	O
.	O
</s>
<s>
For	O
example	O
,	O
an	O
analytic	B-Language
function	I-Language
is	O
the	O
limit	B-Algorithm
of	O
its	O
Taylor	O
series	O
,	O
within	O
its	O
radius	O
of	O
convergence	B-Algorithm
.	O
</s>
<s>
This	O
is	O
known	O
as	O
the	O
harmonic	B-Algorithm
series	I-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
derived	O
from	O
Viète	B-Algorithm
's	I-Algorithm
formula	I-Algorithm
for	O
.	O
</s>
