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MathGroup Archive 2004

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Re: Re: Calculus : limits

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51366] Re: [mg51301] Re: [mg51279] Calculus : limits
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Fri, 15 Oct 2004 02:46:41 -0400 (EDT)
  • Organization: Mathematics & Statistics, Univ. of Mass./Amherst
  • References: <200410120557.BAA19198@smc.vnet.net> <200410141035.GAA14797@smc.vnet.net>
  • Reply-to: murray at math.umass.edu
  • Sender: owner-wri-mathgroup at wolfram.com

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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