Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > comp.soft-sys.math.mathematica > #2051

Re: Limit[f[x], x->a] vs. f[a]. When are they equal?

Show all headers | View raw

On 5/2/2011 9:23 AM, Andrzej Kozlowski wrote:
> On 2 May 2011, at 17:24, Richard Fateman wrote:
>
>> On 5/2/2011 6:49 AM, Andrzej Kozlowski wrote:
>>
>> ...
>>> I guess I would prefer the answer ComplexInfinity but this is not necessary since, for example, Sin[x]/x also returns Indeterminate rather than 1 at 0.
>> I think you mean ... at ComplexInfinity.
>>
>>>   This is the kind od thing any user can decide for himself   by explicitly defining g[ComplexInfinity]=ComplexInfinity.
>>>
>>> There is absolutely nothing here that in any way disagrees with any mathematics I know of.
>> let r[x_]:=Sin[x]/x.
>>
>> Limit[r[x],x->Infinity]  is 0
> That's the right answer of course!
That's  one of the correct ones.  I thought I would just check it.

>
>> Limit[r[x],x->ComplexInfinity]    is unevaluated.
> That's also the right answer.
Well, every answer that comes from Mathematica is by (your) definition, 
right. Because what the program does is what the program is defined to 
do by its program which is by definition right.

Certainly an expression that has different values when it approached in 
the limit from different directions does not "have a limit" and so one 
has a choice:
(a) give an error message
(b) leave the command unevaluated
(c) return a special symbol, like Indeterminate.
(d) maybe something else, like an Interval?

It seems that any time Limit[..] determines that there is no limit, in 
some conventional sense, it returns the Limit[ ..] form unevaluated, and 
so (according to DanL) it doesn't return Indeterminate, but the Limit[ ] 
form.  Except sometimes.
e.g. Limit[Indeterminate^x,x->0]   DOES return Indeterminate.

There is also the question as to what to return in a case in which the 
limit MAY exist but we don't know what it is.  That seems to me to be 
the appropriate place for the unevaluated Limit.   Not so for the case 
in which we have determined that there is no limit.



> The limit does not exist. You can again check that with Mathematica:
>
>   Limit[Sin[x]/x, x ->  ComplexInfinity, Direction ->  I]
>
> Infinity
>
> But
>
> Limit[Sin[x]/x, x ->  ComplexInfinity, Direction ->  1]
>
> Out[108]= 0

Well would you then say that the limit is Indeterminate?  And would that 
be the answer that a mathematician would expect?
I assume that if you asked your students on an exam to answer the 
question "what is the limit of sin(x)/x as x approaches the point at 
complex infinity"  you would not give full credit to students who answer 
"It is: the limit of sin(x)/x as x approaches the point at complex 
infinity" .


> One directional limit is finite while another is infinite. The fore obviously there can be no undirected limit. In fact, if you knew a little more mathematics, that would be obvious to you. Esentially all these "problems" derive from a single source - your poor mathematical background. You sure would have problems passing my analytic functions course.
I suspect that Mathematica would have trouble :)
RJF

> Andrzej Kozlowski
>

Back to comp.soft-sys.math.mathematica | Previous | Next | Find similar | Unroll thread

Thread

Re: Limit[f[x], x->a] vs. f[a]. When are they equal? Richard Fateman <fateman@eecs.berkeley.edu> - 2011-05-03 09:46 +0000

csiph-web