       Re: Calculus : limits

• To: mathgroup at smc.vnet.net
• Subject: [mg51301] Re: [mg51279] Calculus : limits
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 14 Oct 2004 06:35:18 -0400 (EDT)
• References: <200410120557.BAA19198@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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
documentation in version 4 (if not earlier).

Andrzej Kozlowski
Chiba, Japan
http://www.akikoz.net/~andrzej/
http://www.mimuw.edu.pl/~akoz/

```

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