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

**References**:**Calculus : limits***From:*Amir <z64043@netscape.net>

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