Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- To: mathgroup at smc.vnet.net
- Subject: [mg118371] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- From: Richard Fateman <fateman at eecs.berkeley.edu>
- Date: Mon, 25 Apr 2011 07:26:54 -0400 (EDT)
On 4/24/2011 12:32 PM, Daniel Lichtblau wrote:
>
>> (RJF wrote)
>> I poked around some more..
>>
>> Limit[1 - Exp[I x], x -> 0] yields 0
>> Limit[1/x, x->0] yields Infinity
>>
>> Limit[1/(1 - Exp[I x]), x -> 0] yields I*Infinity. ??
>> I would think this would be a place for ComplexInfinity.
> The series expansion would be 1/(1-(1+I*x+...)) = 1/(-I*x+...) --> I*Infinity. This is using the Mathematica limit direction convention (more below) so the 'x' is regarded as positive.
Well, we actually know more about it, I think. From the series
expansion, that is.
First, a bug?
Series[1/(1-Exp[I x]), {x, 0, 4}] returns unevaluated. A bug?
1/(1 - Series[Exp[I x], {x, 0, 4}]), which I would have thought was the
same as the line above, works.
The series starts out as I/x +1/2 -I x/12 + ....
So it seems that if the limit is L, then Re[L] is 1/2 and Im[L] is (by
convention of x approaching zero from + direction) , positive infinity.
So the limit might be 1/2+I Infinity?
>
>> All in Mathematica 7.0
> If I remember correctly we try to return unevaluated in preference to Indeterminate for Limit.
OK, that might work, though sometimes Limit returns Intervals when I
think Indeterminate might be preferable.
I found Limit[Exp[I x],x->Infinity] //N amusing..
> Recall that limits are in general path dependent. The examples you show just above all have clearly defined paths. Per documentation, direction is from positive to negative unless specified otherwise by option, or forced otherwise in approaching an infinity. When the point approached is ComplexInfinity==DirectedInfinity[] we neither are given nor infer a direction. As sine has an essential singularity at (complex) infinity, there is no viable limit nor even a plausible Interval-type result. Ergo, it is unevaluated.
OK; generalizing from reals or complex numbers to infinities is dangerous.
Sometimes the "most" correct answer is partly a matter of opinion.
Thanks
RJF