       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.)
>
> --
> 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

```

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