Re: Re: Re: Re: Calculus : limits

• To: mathgroup at smc.vnet.net
• Subject: [mg51438] Re: [mg51398] Re: [mg51366] Re: [mg51301] Re: [mg51279] Calculus : limits
• From: DrBob <drbob at bigfoot.com>
• Date: Sun, 17 Oct 2004 03:05:49 -0400 (EDT)
• References: <200410160820.EAA23650@smc.vnet.net>
• Reply-to: drbob at bigfoot.com
• Sender: owner-wri-mathgroup at wolfram.com

```>> But, as I am sure Bobby Treat would agree, itought to be under Limit.

You got THAT right.

Bobby

On Sat, 16 Oct 2004 04:20:30 -0400 (EDT), Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:

> Yes, you are right. But, as I am sure Bobby Treat would agree ;-) , it
> ought to be under Limit. The point is that unless you have a reason to
> suspect that Limit by default computes directional limits it is
> unlikely you will look under Direction to find out. But I have to agree
> that I was wrong: the long ago made promise to put this into the
> documentation was kept, even though, in my opinion, not in the most
> natural way.
>
> Andrzej
>
>
> On 15 Oct 2004, at 15:46, Murray Eisenberg wrote:
>
>> 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.
>>>>
>>>> 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
>>
>
>
>
>

--
DrBob at bigfoot.com
www.eclecticdreams.net

```

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