Re: Re: Calculus : limits

*To*: mathgroup at smc.vnet.net*Subject*: [mg51361] Re: [mg51311] Re: Calculus : limits*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Fri, 15 Oct 2004 02:46:15 -0400 (EDT)*References*: <ckfs34$isl$1@smc.vnet.net> <200410141035.GAA14847@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Helen Read wrote: > 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 > > > Unfortunately, Mathematica by default takes the limit from the right, > and does not check to see if it's the same as the limit from the left. > It does not actually do a two-sided limit. In any example where the > one-sided limits are not the same, instead of an error message that > the limit does not exist, Mathematica instead gives you the limit from > the right. Worse, there's nothing in the Help that even tells you that > Limit means "limit from the right" unless you specify the left. > > It will do the one-sided limits correctly if you ask for them separately. > > To find the limit as x->0 from the right: > > Limit[(Abs[Sin[x] - Sin[2 x]])/x, x -> 0, Direction -> -1] > > To find the limit as x->0 from the left: > > Limit[(Abs[Sin[x] - Sin[2 x]])/x, x -> 0, Direction -> 1] > > In effect, > > Limit[(Abs[Sin[x]-Sin[2 x]]) / x, x->0] > > is the same as > > Limit[(Abs[Sin[x] - Sin[2 x]])/x, x -> 0, Direction -> -1] > > and is *not* a two-sided limit. > > (I don't like it either.) > > -- > Helen Read > University of Vermont For an explanation of why the notion of a "two sided limit" makes little sense for a general Limit function, I refer to a prior post to MathGroup: http://forums.wolfram.com/mathgroup/archive/2001/Nov/msg00190.html I tend to agree that the default behavior of Direction->Automatic warrants explicit documentation. Daniel Lichtblau Wolfram Research

**References**:**Re: Calculus : limits***From:*Helen Read <read@math.uvm.edu>