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

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  Re: Limit[f[x], x->a] vs. f[a]. When are they equal? Andrzej Kozlowski <akoz@mimuw.edu.pl> - 2011-05-03 09:44 +0000

#2041 — Re: Limit[f[x], x->a] vs. f[a]. When are they equal?

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!


> Limit[r[x],x->ComplexInfinity]    is unevaluated.

That's also the right answer. 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

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.

Andrzej Kozlowski

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