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.
Please help...
Amir
Try with:
Limit[(Abs[Sin[x]-Sin[2 x]]) / x, x->0,Direction->1]
and you will get the answer as x approaches to x0 from smaller values
(odd limit). Read mathematica help for more informations on Limit[] and
Direction option.
Regards.
Matteo Delfino
> 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
Amir <z64...@netscape.net> wrote in message news:<ckfs34$isl$1...@smc.vnet.net>...
>
> 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.
> Please help...
>
> 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/
In[6]:=
Limit[Abs[Sin[x] - Sin[2*x]]/x, x -> 0, Direction -> 1]
Out[6]=
-1
In[5]:=
Limit[Abs[Sin[x] - Sin[2*x]]/x, x -> 0, Direction -> -1]
Out[5]=
1
Wolfgang
>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.
Yep, the limit from the left is -1, while that from the right is +1.
They can hardly agree in zero. : ]
Ultimately your limit is equivalent to that of Abs[x]/x for x->0
You should load the package Calculus`Limit` to compute limits with
discontinuos functions such as Abs.
(But don't forget, as you've just done, to double check the result you get).
cheers,
Peltio
Invalid address in reply-to. Crafty demunging required to mail me.
In[1]:=Limit[Abs[Sin[x] - Sin[2x]]/x, x -> 0, Direction -> -1]
Out[1]=1
In[2]:=Limit[Abs[Sin[x] - Sin[2x]]/x, x -> 0, Direction -> 1]
Out[2]=-1
v.a.
--
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Vous aussi inscrivez-vous sans plus tarder!!
Message posté à partir de http://www.gyptis.org, BBS actif depuis 1995.
f[x_] = (Sin[x] - Sin[2*x])/x;
Limit[f[x], x -> 0]
-1
Limit[Abs[f[x]], x -> 0]
1
Plot[f[x], {x, -1, 1}]
A look at the Series representations makes the answer very clear:
Series[Sin[x], {x, 0, 5}]
Series[Sin[2*x], {x, 0, 5}]
(%% - %)/x
SeriesData[x, 0, {1, 0, -1/6, 0, 1/120}, 1, 6, 1]
SeriesData[x, 0, {2, 0, -4/3, 0, 4/15}, 1, 6, 1]
SeriesData[x, 0, {-1, 0, 7/6, 0, -31/120}, 0, 5, 1]
Or, in even simpler terms, when x is close to 0, Sin[x] is close to x and Sin[2x] is close to 2x, so their difference is close to -x. Divide by x, and that's close to -1. Take Abs and you get 1.
Bobby
On Tue, 12 Oct 2004 01:57:42 -0400 (EDT), Amir <z64...@netscape.net> 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.
> Please help...
>
> Amir
>
>
>
>
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.
>>Please help...
>>
>>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 mur...@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
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
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
>>>
You got THAT right.
Bobby