Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- To: mathgroup at smc.vnet.net
- Subject: [mg118418] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- From: CPE Bach <hisicon at gmail.com>
- Date: Wed, 27 Apr 2011 05:40:00 -0400 (EDT)
This is more a math problem, rather than a Mathematica question.
Sent from my iPhone
On Apr 26, 2011, at 6:50 AM, Stefan <wutchamacallit27 at gmail.com> wrote:
> On Apr 26, 4:43 am, Richard Fateman <fate... at cs.berkeley.edu> wrote:
>> On 4/25/2011 4:29 AM, Andrzej Kozlowski wrote:
>> ....
>>
>>
>>
>>> I forgot to add that the function Sin is certainly not continuous at infinity (Sin[1/z]
>>
>> has an essential singularity at 0)
>>
>> so there is no reason why it's value at there should agree with its limit.
>>
>> In fact, it definitely should not do so.
>>
>>
>>
>>> Andrzej Kozlowski
>>
>> Hm. continuity (excerpt from wikipedia, but whatever)
>>
>> The limit of f(x) as x approaches c through domain of f does exist and
>> is equal to f(c); in mathematical notation, \lim_{x \to c}{f(x)} = f(c).
>> If the point c in the domain of f is not a limit point of the domain,
>> then this condition is vacuously true, since x cannot approach c through
>> values not equal c. Thus, for example, every function whose domain is
>> the set of all integers is continuous.
>>
>> ........
>> There is now a kind of semantic gap, it seems to me. There are, perhaps,
>> several gaps.
>>
>> One gap is between the mathematical concept of limit and the result of
>> computing Limit[] in Mathematica. Often they corresponds. Sometimes
>> they differ. I see this difference appearing when the result in
>> Mathematica is an Interval, and sometimes when the result is
>> ComplexInfinity or its relatives. I do not believe that the
>> mathematical result 1/2+ i*(infinity) is the same as Mathematica's
>> I*Infinity or ComplexInfinity. See below for a proof.
>>
>> Another gap is around special symbolic values that might be used as
>> limit points but which have no neighborhoods. Look at the definition
>> above. Is there some set of Mathematica values x1, x2, x3, ...., xn, ...
>> all not equal to c, such that their distance from c becomes smaller,
>> when c = Infinity? Apparently not, since Abs[x-Infinity] is Infinity for
>> all x.
>>
>> So we fall into the vacuously true clause, in which f[c] is the limit.
>>
>> This is, I think, a consequence of introducing oddball pseudo-real
>> objects into your computer system (notions like indeterminate,
>> intervals, infinity).
>>
>> It's possible I've overlooked something here and another definition of
>> continuity more suitable for "hyperreals" or some computer-algebra
>> topology fixes this all up.
>>
>> Here's a fun thing to do with our function f[x_]:=1/(1-Exp[I x]) .
>>
>> try Plot[Re[f[x]],{x,0.01,20}]
>> On my system it look like a plot of y=0. A bug (Mathematica 7)?
>>
>> And yet Re[f[0.5]] is 0.5, not zero. same for Re[f[0.01]].
>>
>> Next try Plot[f[x]],{x,0.01,200}], to see some bubbles and a large oddly
>> positioned spike between 70 and 80.
>>
>> In fact, Re[f[x]] should always be 1/2.
>>
>> As for Mathematica's return of Limit[f[x],x->0] as I*Infinity, I'm
>> pretty sure that 1/2+I*Infinity is a better answer. I suspect it is a
>> mistake to "simplify" that to I*Infinity. Here's why.
>>
>> Limit[f[x]-I/x,x->0] comes out 1/2.
>>
>> My answer preserves the identity lim(A+B) = lim(A)+lim(B).
>>
>> Mathematica's does not. [This identity holds if lim(A) and lim(B) both
>> exist, which apparently they do in Mathematica.)
>>
>> Back to Andrzej's note.
>>
>> Sin(1/z) as z->0 is not the same as Sin[z] as z->ComplexInfinity.
>>
>> 1/z as z->0 has a neighborhood. There is no neighborhood around
>> ComplexInfinity.
>>
>> (I am not, incidentally, proposing that I have a complete solution to
>> these issues.)
>
> Richard,
>
> Intrigued by your claims, I tried the same plots and did not see what
> you describe. I believe you may have overlooked the labels of your
> axes in these plots, since Mathematica doesnt always have the x and y
> axes meet at the origin. In the first plot you mention, Plot[Re[f[x]],
> {x, 0.01, 20}], the axes meet at (x,y) = (0,0.5) and the function is
> constant = 0.5.
> In the extended plot Plot[Re[f[x]], {x, 0.01, 200}], I do see the
> bumps you mention, and I have no idea why that happens, I believe it
> to be a question separate from the discussions here. Though I will
> note that the axes on my plot ranged from 0.5 to 0.5 in increments of
> zero! This seems to be some strange manifestation of numerical
> precision in Plot, rather than a problem with Mathematica evaluating
> these functions. Zooming in on the largest bump in that graph, between
> 70 and 80, I actually found that the bump appeared to extend upward,
> where the first graph showed it going down (though all still on a
> seemingly infinitesimal range 0.5 to 0.5. This seems like an
> interesting question regarding the numerical methods of Plot which
> deserves its own thread.
> Finally, with regard to your argument about the limit. I both agree
> with your reasoning, but disagree that it is a mistake to say that the
> limit is I*Infinity. I think the issue may be more context sensitive.
> While it might be important that your function has real part -> 1/2,
> it is also the case that when compared to an imaginary part ->
> Infinity, the real part is insignificant. Consider the phase of the
> limit, Limit[Arg[f[x]],x->0], you would agree this is ArcTan[Infinity/
> (1/2)] = Pi/2, and so I*Infinity is an appropriate answer. If you are
> interested in the limit of the real part, then use Limit[Re[f[x]],x-
>> 0] which does indeed give you 1/2.
>
> -Stefan S
>