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MathGroup Archive 2004

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Re: Re: Re: Calculus : limits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51398] Re: [mg51366] Re: [mg51301] Re: [mg51279] Calculus : limits
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 16 Oct 2004 04:20:30 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Yes, you are right. But, as I am sure Bobby Treat would agree ;-) , it 
ought to be under Limit. The point is that unless you have a reason to 
suspect that Limit by default computes directional limits it is 
unlikely you will look under Direction to find out. But I have to agree 
that I was wrong: the long ago made promise to put this into the 
documentation was kept, even though, in my opinion, not in the most 
natural way.

Andrzej


On 15 Oct 2004, at 15:46, Murray Eisenberg wrote:

> The documentation is there in the front end (at least in Mathematica
> 5.0.1), just not in The Mathematica Book:
>
>    Options[Limit]
> {Analytic -> False, Assumptions :> $Assumptions, Direction -> 
> Automatic}
>
>    ?Direction
> Direction is an option for Limit. Limit[expr, x -> x0, Direction -> 1]
> computes the limit as x approaches x0 from smaller values. Limit[expr, 
> x
> -> x0, Direction -> -1] computes the limit as x approaches x0 from
> larger values. Direction -> Automatic uses Direction -> -1 except for
> limits at Infinity, where it is equivalent to Direction -> 1.
>
>
> Andrzej Kozlowski wrote:
>> On 12 Oct 2004, at 14:57, Amir wrote:
>>
>>
>>> Hi,
>>>
>>> I'd like to find the limit of
>>> Limit[(Abs[Sin[x]-Sin[2 x]]) / x, x->0]
>>>
>>> I use Mathematica v.5. I get the wrong (??) answer : 1
>>>
>>> While I try to display the graph of this function by using "Plot", it
>>> seems that there is no limit at the point x=0.
>>> Please help...
>>>
>>> Amir
>>>
>>
>>
>> Mathematica's answer is correct but ... Limit always computes
>> directional limits. Thus:
>>
>>
>> Limit[Abs[Sin[x] - Sin[2*x]]/x, x -> 0, Direction -> -1]
>>
>> 1
>>
>> but
>>
>> Limit[Abs[Sin[x] - Sin[2*x]]/x, x -> 0, Direction -> 1]
>>
>> -1
>>
>> So the limits as x goes to 0 form above and form below are different
>> and thus "there is n limit'.
>>
>> Also, as you see by default Limit computes "from above". However, I
>> still can't find this clearly documented in version 5, even though I
>> remeber myself (and others) complaining about this lack of
>> documentation in version 4 (if not earlier).
>>
>>
>>
>>
>> Andrzej Kozlowski
>> Chiba, Japan
>> http://www.akikoz.net/~andrzej/
>> http://www.mimuw.edu.pl/~akoz/
>>
>>
>>
>
> -- 
> Murray Eisenberg                     murray at math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower      phone 413 549-1020 (H)
> University of Massachusetts                413 545-2859 (W)
> 710 North Pleasant Street            fax   413 545-1801
> Amherst, MA 01003-9305
>


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