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MathGroup Archive 2007

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Re: Bug with Limit?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74178] Re: Bug with Limit?
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Tue, 13 Mar 2007 05:09:36 -0500 (EST)
  • References: <200703021134.GAA03672@smc.vnet.net><esb4eh$3ql$1@smc.vnet.net>

> Making the documentation clear is a problem for the designers
> of Mathematica.

Everything is clear and "out there"...(at least for me!)


In[2]:=
Options[Limit]
(Information[Evaluate[#1[[1]]]] & ) /@ %;

Out[2]=
{Analytic -> False, Assumptions :> $Assumptions, Direction ->
Automatic}

>From In[2]:=
"Analytic is an option for Limit and Series. With Analytic -> True,
unrecognized functions are treated as analytic, and processed using
Taylor series expansions; with Analytic -> False, Taylor series are
not used unless the function is recognized as analytic."
>From In[2]:=
Attributes[Analytic] = {Protected}

>From In[2]:=
"Assumptions is an option for functions such as Simplify, Refine and
Integrate which specifies default assumptions to be made about
symbolic quantities."
>From In[2]:=
Attributes[Assumptions] = {Protected}

>From In[2]:=
"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."
>From In[2]:=
Attributes[Direction] = {Protected}

Execute also the following command:

In[5]:=
FrontEndExecute[{HelpBrowserLookup["MainBook", "3.6.9", "9.16"]}]


BTW...

There are a great many books regarding calculus and Mathematica.
http://store.wolfram.com/catalog/books/calculus.html

The following links is also of particular interest.

http://support.wolfram.com/mathematica/kernel/Symbols/System/Direction.html
http://support.wolfram.com/mathematica/kernel/Symbols/System/Analytic.html
http://library.wolfram.com/infocenter/Courseware/4707/
http://library.wolfram.com/infocenter/TechNotes/82/


Last but not least...

In[7]:=
Plot[Exp[1/x], {x, -2, 2.1}, PlotRange -> {0, 5}]

In[10]:=
(Limit[Exp[1/x], x -> 0, Direction -> #1] & ) /@ {1, -1}
Out[1]=
{0, Infinity}

In[13]:=
(Limit[Exp[1/(I*x)], x -> 0, Direction -> #1] & ) /@ {-I, I}
Out[13]=
{0, Infinity}

In[15]:=
(Limit[Sin[1/x], x -> 0, Direction -> #1] & ) /@ {Plus[1], -1}
(Limit[Sin[1/x], x -> 0, Direction -> #1] & ) /@ {Plus[I], -I}
Out[15]=
{Interval[{-1, 1}], Interval[{-1, 1}]}
Out[16]=
{DirectedInfinity[I], DirectedInfinity[-I]}

In[17]:=
Limit[(f[z + =E5] - f[z])/=E5, =E5 -> 0]
Out[17]=
Limit[(-f[z] + f[z + =E5])/=E5, =E5 -> 0]

In[18]:=
Limit[(f[z + =E5] - f[z])/=E5, =E5 -> 0, Analytic -> True]
Out[18]=
Derivative[1][f][z]


Kind Regards
Dimitris


=CF/=C7 Eric Smith =DD=E3=F1=E1=F8=E5:
> Andrzej Kozlowski <akoz at mimuw.edu.pl> writes:
> > There is no reason why any compromises should be made in
> > design of Mathematica to deal with problems of students who do not
> > have yet acquired sufficient understanding of the mathematical
> > concepts involved.
>
> No one is asking for that.  But it would be nice if the Help Browser
> page for the Limit function made it explicit that if a direction
> is not specified, a default direction is used.  Otherwise there
> will be people who think that omitting the direction indicates that
> a non-directional limit is being requested.
>
> Yes, if one digs into the documentation far enough, the information
> is there.  But this is significant enough to merit mention on the
> Limit page.
>
> > In my opinion, this is a problem for educators and
> > not for the designers of Mathematica.
>
> Making the documentation clear is a problem for the designers
> of Mathematica.



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