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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>