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Re: Calculus : limits

• To: mathgroup at smc.vnet.net
• Subject: [mg51311] Re: Calculus : limits
• From: Helen Read <read at math.uvm.edu>
• Date: Thu, 14 Oct 2004 06:35:43 -0400 (EDT)
• Organization: EM Computer Facility, The University of Vermont
• References: <ckfs34$isl$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

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

• Follow-Ups:
  • Re: Re: Calculus : limits
    • From: Daniel Lichtblau <danl@wolfram.com>