<s>
In	O
calculus	O
and	O
other	O
branches	O
of	O
mathematical	O
analysis	O
,	O
limits	B-Algorithm
involving	O
an	O
algebraic	O
combination	O
of	O
functions	O
in	O
an	O
independent	O
variable	O
may	O
often	O
be	O
evaluated	O
by	O
replacing	O
these	O
functions	O
by	O
their	O
limits	B-Algorithm
;	O
if	O
the	O
expression	O
obtained	O
after	O
this	O
substitution	O
does	O
not	O
provide	O
sufficient	O
information	O
to	O
determine	O
the	O
original	O
limit	B-Algorithm
,	O
then	O
the	O
expression	O
is	O
called	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
More	O
specifically	O
,	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
is	O
a	O
mathematical	O
expression	O
involving	O
at	O
most	O
two	O
of	O
,	O
or	O
,	O
obtained	O
by	O
applying	O
the	O
algebraic	O
limit	B-Algorithm
theorem	O
in	O
the	O
process	O
of	O
attempting	O
to	O
determine	O
a	O
limit	B-Algorithm
,	O
which	O
fails	O
to	O
restrict	O
that	O
limit	B-Algorithm
to	O
one	O
specific	O
value	O
or	O
infinity	O
,	O
and	O
thus	O
does	O
not	O
determine	O
the	O
limit	B-Algorithm
being	O
sought	O
.	O
</s>
<s>
A	O
limit	B-Algorithm
confirmed	O
to	O
be	O
infinity	O
is	O
not	O
indeterminate	O
since	O
it	O
has	O
been	O
determined	O
to	O
have	O
a	O
specific	O
value	O
(	O
infinity	O
)	O
.	O
</s>
<s>
There	O
are	O
seven	O
indeterminate	B-Algorithm
forms	I-Algorithm
which	O
are	O
typically	O
considered	O
in	O
the	O
literature	O
:	O
</s>
<s>
The	O
most	O
common	O
example	O
of	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
occurs	O
when	O
determining	O
the	O
limit	B-Algorithm
of	O
the	O
ratio	O
of	O
two	O
functions	O
,	O
in	O
which	O
both	O
of	O
these	O
functions	O
tend	O
to	O
zero	O
in	O
the	O
limit	B-Algorithm
,	O
and	O
is	O
referred	O
to	O
as	O
"	O
the	O
indeterminate	B-Algorithm
form	I-Algorithm
"	O
.	O
</s>
<s>
In	O
each	O
case	O
,	O
if	O
the	O
limits	B-Algorithm
of	O
the	O
numerator	O
and	O
denominator	O
are	O
substituted	O
,	O
the	O
resulting	O
expression	O
is	O
,	O
which	O
is	O
undefined	O
.	O
</s>
<s>
In	O
a	O
loose	O
manner	O
of	O
speaking	O
,	O
can	O
take	O
on	O
the	O
values	O
,	O
,	O
or	O
,	O
and	O
it	O
is	O
easy	O
to	O
construct	O
similar	O
examples	O
for	O
which	O
the	O
limit	B-Algorithm
is	O
any	O
particular	O
value	O
.	O
</s>
<s>
Not	O
every	O
undefined	O
algebraic	O
expression	O
corresponds	O
to	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
expression	O
is	O
undefined	O
as	O
a	O
real	O
number	O
but	O
does	O
not	O
correspond	O
to	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
;	O
any	O
defined	O
limit	B-Algorithm
that	O
gives	O
rise	O
to	O
this	O
form	O
will	O
diverge	O
to	O
infinity	O
.	O
</s>
<s>
An	O
expression	O
that	O
arises	O
by	O
ways	O
other	O
than	O
applying	O
the	O
algebraic	O
limit	B-Algorithm
theorem	O
may	O
have	O
the	O
same	O
form	O
of	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
However	O
it	O
is	O
not	O
appropriate	O
to	O
call	O
an	O
expression	O
"	O
indeterminate	B-Algorithm
form	I-Algorithm
"	O
if	O
the	O
expression	O
is	O
made	O
outside	O
the	O
context	O
of	O
determining	O
limits	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
which	O
arises	O
from	O
substituting	O
for	O
in	O
the	O
equation	O
is	O
not	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
since	O
this	O
expression	O
is	O
not	O
made	O
in	O
the	O
determination	O
of	O
a	O
limit	B-Algorithm
(	O
it	O
is	O
in	O
fact	O
undefined	O
as	O
division	B-Algorithm
by	I-Algorithm
zero	I-Algorithm
)	O
.	O
</s>
<s>
For	O
more	O
,	O
see	O
the	O
article	O
Zero	B-Algorithm
to	I-Algorithm
the	I-Algorithm
power	I-Algorithm
of	I-Algorithm
zero	I-Algorithm
.	O
</s>
<s>
Note	O
that	O
and	O
other	O
expressions	O
involving	O
infinity	O
are	O
not	O
indeterminate	B-Algorithm
forms	I-Algorithm
.	O
</s>
<s>
The	O
indeterminate	B-Algorithm
form	I-Algorithm
is	O
particularly	O
common	O
in	O
calculus	O
,	O
because	O
it	O
often	O
arises	O
in	O
the	O
evaluation	O
of	O
derivatives	B-Algorithm
using	O
their	O
definition	O
in	O
terms	O
of	O
limit	B-Algorithm
.	O
</s>
<s>
This	O
is	O
enough	O
to	O
show	O
that	O
is	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
Direct	O
substitution	O
of	O
the	O
number	O
that	O
approaches	O
into	O
any	O
of	O
these	O
expressions	O
shows	O
that	O
these	O
are	O
examples	O
correspond	O
to	O
the	O
indeterminate	B-Algorithm
form	I-Algorithm
,	O
but	O
these	O
limits	B-Algorithm
can	O
assume	O
many	O
different	O
values	O
.	O
</s>
<s>
Any	O
desired	O
value	O
can	O
be	O
obtained	O
for	O
this	O
indeterminate	B-Algorithm
form	I-Algorithm
as	O
follows	O
:	O
</s>
<s>
The	O
following	O
limits	B-Algorithm
illustrate	O
that	O
the	O
expression	O
is	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
:	O
</s>
<s>
If	O
the	O
functions	O
and	O
are	O
analytic	B-Language
at	O
,	O
and	O
is	O
positive	O
for	O
sufficiently	O
close	O
(	O
but	O
not	O
equal	O
)	O
to	O
,	O
then	O
the	O
limit	B-Algorithm
of	O
will	O
be	O
.	O
</s>
<s>
Otherwise	O
,	O
use	O
the	O
transformation	O
in	O
the	O
table	O
below	O
to	O
evaluate	O
the	O
limit	B-Algorithm
.	O
</s>
<s>
The	O
expression	O
is	O
not	O
commonly	O
regarded	O
as	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
,	O
because	O
if	O
the	O
limit	B-Algorithm
of	O
exists	O
then	O
there	O
is	O
no	O
ambiguity	O
as	O
to	O
its	O
value	O
,	O
as	O
it	O
always	O
diverges	O
.	O
</s>
<s>
The	O
limit	B-Algorithm
fails	O
to	O
exist	O
.	O
</s>
<s>
In	O
each	O
case	O
the	O
absolute	O
value	O
approaches	O
,	O
and	O
so	O
the	O
quotient	O
must	O
diverge	O
,	O
in	O
the	O
sense	O
of	O
the	O
extended	O
real	O
numbers	O
(	O
in	O
the	O
framework	O
of	O
the	O
projectively	O
extended	O
real	O
line	O
,	O
the	O
limit	B-Algorithm
is	O
the	O
unsigned	O
infinity	O
in	O
all	O
three	O
cases	O
)	O
.	O
</s>
<s>
Similarly	O
,	O
any	O
expression	O
of	O
the	O
form	O
with	O
(	O
including	O
and	O
)	O
is	O
not	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
,	O
since	O
a	O
quotient	O
giving	O
rise	O
to	O
such	O
an	O
expression	O
will	O
always	O
diverge	O
.	O
</s>
<s>
The	O
expression	O
is	O
not	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
The	O
expression	O
obtained	O
from	O
considering	O
gives	O
the	O
limit	B-Algorithm
,	O
provided	O
that	O
remains	O
nonnegative	O
as	O
approaches	O
.	O
</s>
<s>
The	O
expression	O
is	O
similarly	O
equivalent	O
to	O
;	O
if	O
as	O
approaches	O
,	O
the	O
limit	B-Algorithm
comes	O
out	O
as	O
.	O
</s>
<s>
The	O
adjective	O
indeterminate	O
does	O
not	O
imply	O
that	O
the	O
limit	B-Algorithm
does	O
not	O
exist	O
,	O
as	O
many	O
of	O
the	O
examples	O
above	O
show	O
.	O
</s>
<s>
In	O
many	O
cases	O
,	O
algebraic	O
elimination	O
,	O
Hôpital	O
's	O
rule	O
,	O
or	O
other	O
methods	O
can	O
be	O
used	O
to	O
manipulate	O
the	O
expression	O
so	O
that	O
the	O
limit	B-Algorithm
can	O
be	O
evaluated	O
.	O
</s>
<s>
When	O
two	O
variables	O
and	O
converge	B-Algorithm
to	O
zero	O
at	O
the	O
same	O
limit	B-Algorithm
point	O
and	O
,	O
they	O
are	O
called	O
equivalent	O
infinitesimal	O
(	O
equiv	O
.	O
</s>
<s>
For	O
the	O
evaluation	O
of	O
the	O
indeterminate	B-Algorithm
form	I-Algorithm
,	O
one	O
can	O
make	O
use	O
of	O
the	O
following	O
facts	O
about	O
equivalent	O
infinitesimals	O
(	O
e.g.	O
,	O
if	O
x	O
becomes	O
closer	O
to	O
zero	O
)	O
:	O
</s>
<s>
Hôpital	O
's	O
rule	O
is	O
a	O
general	O
method	O
for	O
evaluating	O
the	O
indeterminate	B-Algorithm
forms	I-Algorithm
and	O
.	O
</s>
<s>
where	O
and	O
are	O
the	O
derivatives	B-Algorithm
of	O
and	O
.	O
</s>
<s>
(	O
Note	O
that	O
this	O
rule	O
does	O
not	O
apply	O
to	O
expressions	O
,	O
,	O
and	O
so	O
on	O
,	O
as	O
these	O
expressions	O
are	O
not	O
indeterminate	B-Algorithm
forms	I-Algorithm
.	O
)	O
</s>
<s>
These	O
derivatives	B-Algorithm
will	O
allow	O
one	O
to	O
perform	O
algebraic	O
simplification	O
and	O
eventually	O
evaluate	O
the	O
limit	B-Algorithm
.	O
</s>
<s>
Hôpital	O
's	O
rule	O
can	O
also	O
be	O
applied	O
to	O
other	O
indeterminate	B-Algorithm
forms	I-Algorithm
,	O
using	O
first	O
an	O
appropriate	O
algebraic	O
transformation	O
.	O
</s>
<s>
The	O
following	O
table	O
lists	O
the	O
most	O
common	O
indeterminate	B-Algorithm
forms	I-Algorithm
and	O
the	O
transformations	O
for	O
applying	O
Hôpital	O
's	O
rule	O
.	O
</s>
