Re: Re: One to the power Infinity
- To: mathgroup at smc.vnet.net
- Subject: [mg36036] Re: [mg36001] Re: One to the power Infinity
- From: "Jonathan Rockmann" <MTheory at msn.com>
- Date: Tue, 13 Aug 2002 05:22:59 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Selwyn,
In the plot of Abs[(1-x)^(1/x^2)] that you mention, are you adjusting
PlotRange to about PlotRange->{-4,5}?
Jonathan
mtheory at msn.com
----- Original Message -----
From: Selwyn Hollis
To: mathgroup at smc.vnet.net
Subject: [mg36036] [mg36001] Re: One to the power Infinity
David W. Cantrell wrote:
> Selwyn Hollis <slhollis at earthlink.net> wrote:
>
>>Because it depends on how you get there. For instance,
>>Limit[(1-x)^(1/x), x->0] gives 1/E, while Limit[(1-x)^(1/x^2), x->0]
>>gives 0.
>
>
> You are correct. However, that last limit claim is somewhat deceptive.
> Based on it, one might conclude, incorrectly, that the direction of
> approach to 0 is immaterial. Note that, although
> Limit[(1-x)^(1/x^2), x->0, Direction-> -1] yields 0,
> Limit[(1-x)^(1/x^2), x->0, Direction-> +1] yields Infinity.
> Thus, when Mathematica says that Limit[(1-x)^(1/x^2), x->0] is 0, it is
> making a hidden assumption of direction of approach.
Actually it seems you've discovered a bug in Limit. If we allow complex
values, then Limit[(1-x)^(1/x^2), x->0, Direction-> +1] should be 0,
which is evident from the graph of Abs[(1-x)^(1/x^2)]. Mathematica gives
Infinity for Limit[Abs[(1-x)^(1/x^2)], x->0, Direction-> +1] as well. If
we don't allow complex values, then Limit[(1-x)^(1/x^2), x->0,
Direction-> +1] can't exist at all (even as Infinity or -Infinity).
>>In fact, you can construct similar examples in which
>>"1^Infinity" = anything you like.
>
> Well, yes, as long as you don't like negative values.
>
> David
Some of my favorite numbers are negative, such as
(1 + x I)^(I Pi/Log[1 + x I]) = -1
Note that this has the form 1^Infinity as x->0.
>
>
>>Matthias.Bode at oppenheim.de wrote:
>>
>>>1^\[Infinity] => Indeterminate, unexpected. Naively expected: 1.
>>>
>>>For which reason(s) is 1^\[Infinity] defined as Indeterminate?
>>
>