Re: Re: Re: Re: Calculus : limits

*To*: mathgroup at smc.vnet.net*Subject*: [mg51438] Re: [mg51398] Re: [mg51366] Re: [mg51301] Re: [mg51279] Calculus : limits*From*: DrBob <drbob at bigfoot.com>*Date*: Sun, 17 Oct 2004 03:05:49 -0400 (EDT)*References*: <200410160820.EAA23650@smc.vnet.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

>> But, as I am sure Bobby Treat would agree, itought to be under Limit. You got THAT right. Bobby On Sat, 16 Oct 2004 04:20:30 -0400 (EDT), Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: > 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 >> > > > > -- DrBob at bigfoot.com www.eclecticdreams.net

**References**:**Re: Re: Re: Calculus : limits***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>