Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- To: mathgroup at smc.vnet.net
- Subject: [mg118582] Re: Limit[f[x], x->a] vs. f[a]. When are they equal?
- From: Richard Fateman <fateman at eecs.berkeley.edu>
- Date: Tue, 3 May 2011 05:48:31 -0400 (EDT)
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
Limit[r[x],x->ComplexInfinity] is unevaluated.
r[x] /. x-> Infinity is 0
r[x] /. x->ComplexInfinity is Indeterminate
r[Infinity] is Interval[{0,0}] (!!!! version 7.0, anyway)
FullSimplify[%] leaves that unchanged, but
Interval[{0,0}]==0 is True.
Interval[{0,0}]===0 is False
Interval[0]==0 is True
Indeterminate === Indeterminate is True
Indeterminate == Indeterminate is unevaluated.
This is kind of interesting, to have two things which are identical but
not equal.
It is difficult to deal with something that is indeterminate, when it is
actually represented in the computer by something determinate, namely
the symbol Indeterminate. Having a indexed set of Indeterminates (as I
have proposed), solves some of these problems.
It depends on what mathematics you know.
RJF