<s>
In	O
mathematics	O
,	O
division	B-Algorithm
by	I-Algorithm
infinity	I-Algorithm
is	O
division	O
where	O
the	O
divisor	O
(	O
denominator	O
)	O
is	O
infinity	B-Application
.	O
</s>
<s>
In	O
ordinary	O
arithmetic	O
,	O
this	O
does	O
not	O
have	O
a	O
well-defined	O
meaning	O
,	O
since	O
infinity	B-Application
is	O
a	O
mathematical	O
concept	O
that	O
does	O
not	O
correspond	O
to	O
a	O
specific	O
number	O
,	O
and	O
moreover	O
,	O
there	O
is	O
no	O
nonzero	O
real	O
number	O
that	O
,	O
when	O
added	O
to	O
itself	O
an	O
infinite	O
number	O
of	O
times	O
,	O
gives	O
a	O
finite	O
number	O
.	O
</s>
<s>
However	O
,	O
"	O
dividing	O
by	O
infinity	B-Application
"	O
can	O
be	O
given	O
meaning	O
as	O
an	O
informal	O
way	O
of	O
expressing	O
the	O
limit	B-Algorithm
of	O
dividing	O
a	O
number	O
by	O
larger	O
and	O
larger	O
divisors	O
.	O
</s>
<s>
For	O
example	O
,	O
on	O
the	O
extended	O
real	O
number	O
line	O
,	O
dividing	O
any	O
real	O
number	O
by	O
infinity	B-Application
yields	O
zero	O
,	O
while	O
in	O
the	O
surreal	O
number	O
system	O
,	O
dividing	O
1	O
by	O
the	O
infinite	O
number	O
yields	O
the	O
infinitesimal	O
number	O
.	O
</s>
<s>
In	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
,	O
any	O
finite	O
number	O
divided	O
by	O
is	O
equal	O
to	O
positive	B-Algorithm
or	I-Algorithm
negative	I-Algorithm
zero	I-Algorithm
if	O
the	O
numerator	O
is	O
finite	O
.	O
</s>
<s>
Otherwise	O
,	O
the	O
result	O
is	O
NaN	O
.	O
</s>
<s>
The	O
challenges	O
of	O
providing	O
a	O
rigorous	O
meaning	O
of	O
"	O
division	B-Algorithm
by	I-Algorithm
infinity	I-Algorithm
"	O
are	O
analogous	O
to	O
those	O
of	O
defining	O
division	B-Algorithm
by	I-Algorithm
zero	I-Algorithm
.	O
</s>
<s>
As	O
infinity	B-Application
is	O
difficult	O
to	O
deal	O
with	O
for	O
most	O
calculators	O
and	O
computers	O
many	O
do	O
not	O
have	O
a	O
formal	O
way	O
of	O
computing	O
division	B-Algorithm
by	I-Algorithm
infinity	I-Algorithm
.	O
</s>
<s>
Calculators	O
such	O
as	O
the	O
TI-84	B-Device
and	O
most	O
household	O
calculators	O
do	O
not	O
have	O
an	O
infinity	B-Application
button	O
so	O
it	O
is	O
impossible	O
to	O
type	O
into	O
the	O
calculator	O
'	O
x	O
divided	O
by	O
infinity	B-Application
 '	O
so	O
instead	O
users	O
can	O
type	O
a	O
large	O
number	O
such	O
as	O
"	O
1e99	O
"	O
(	O
)	O
or	O
"	O
-1e99	O
"	O
.	O
</s>
<s>
In	O
the	O
Wolfram	B-Language
language	I-Language
,	O
dividing	O
an	O
integer	O
by	O
infinity	B-Application
will	O
result	O
in	O
the	O
result	O
0	O
.	O
</s>
<s>
Also	O
,	O
in	O
some	O
calculators	O
such	O
as	O
the	O
TI-Nspire	B-Device
,	O
1	O
divided	O
by	O
infinity	B-Application
can	O
be	O
evaluated	O
as	O
0	O
.	O
</s>
<s>
Taking	O
the	O
limit	B-Algorithm
of	O
these	O
Riemann	O
sums	O
,	O
in	O
which	O
the	O
sections	O
can	O
be	O
heuristically	O
regarded	O
as	O
"	O
infinitely	O
thin	O
"	O
,	O
gives	O
the	O
definite	O
integral	O
of	O
the	O
function	O
over	O
the	O
prescribed	O
interval	O
.	O
</s>
<s>
Conceptually	O
this	O
results	O
in	O
dividing	O
the	O
interval	O
by	O
infinity	B-Application
to	O
result	O
in	O
infinitely	O
small	O
pieces	O
.	O
</s>
<s>
On	O
a	O
different	O
note	O
when	O
taking	O
an	O
integral	O
where	O
one	O
of	O
the	O
boundaries	O
is	O
infinity	B-Application
this	O
is	O
defined	O
as	O
an	O
improper	O
integral	O
.	O
</s>
<s>
To	O
determine	O
this	O
one	O
would	O
take	O
the	O
limit	B-Algorithm
as	O
a	O
variable	O
a	O
approaches	O
infinity	B-Application
substituting	O
a	O
in	O
for	O
the	O
infinity	B-Application
sign	O
.	O
</s>
<s>
This	O
would	O
then	O
allow	O
the	O
integral	O
to	O
be	O
evaluated	O
and	O
then	O
the	O
limit	B-Algorithm
would	O
be	O
taken	O
.	O
</s>
<s>
In	O
many	O
cases	O
evaluating	O
this	O
would	O
result	O
in	O
a	O
term	O
being	O
divided	O
by	O
infinity	B-Application
.	O
</s>
<s>
This	O
allows	O
for	O
the	O
integral	O
to	O
be	O
assumed	O
to	O
converge	B-Algorithm
meaning	O
a	O
finite	O
answer	O
can	O
be	O
determined	O
from	O
the	O
integral	O
using	O
this	O
assumption	O
.	O
</s>
<s>
When	O
given	O
a	O
ratio	O
between	O
two	O
functions	O
,	O
the	O
limit	B-Algorithm
of	O
this	O
ratio	O
can	O
be	O
evaluated	O
by	O
computing	O
the	O
limit	B-Algorithm
of	O
each	O
function	O
separately	O
.	O
</s>
<s>
Where	O
the	O
limit	B-Algorithm
of	O
the	O
function	O
in	O
the	O
denominator	O
is	O
infinity	B-Application
,	O
and	O
the	O
numerator	O
does	O
not	O
allow	O
the	O
ratio	O
to	O
be	O
well	O
determined	O
,	O
the	O
limit	B-Algorithm
of	O
the	O
ratio	O
is	O
said	O
to	O
be	O
of	O
indeterminate	O
form	O
.	O
</s>
<s>
Using	O
Hôpital	O
's	O
rule	O
to	O
evaluate	O
limits	O
of	O
fractions	O
where	O
the	O
denominator	O
tends	O
towards	O
infinity	B-Application
can	O
produce	O
results	O
other	O
than	O
0	O
.	O
</s>
<s>
This	O
means	O
that	O
,	O
when	O
using	O
limits	O
to	O
give	O
meaning	O
to	O
division	B-Algorithm
by	I-Algorithm
infinity	I-Algorithm
,	O
the	O
result	O
of	O
"	O
dividing	O
by	O
infinity	B-Application
"	O
does	O
not	O
always	O
equal	O
0	O
.	O
</s>
