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MathGroup Archive 2007

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Re: Normal for Limit : Example

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74378] Re: Normal for Limit : Example
  • From: "dimitris" <dimmechan at yahoo.com>
  • Date: Mon, 19 Mar 2007 21:57:31 -0500 (EST)
  • References: <etld77$o39$1@smc.vnet.net>

Hi Ajit.

Could you tell us in what particular post of Andrzej are you refered
to?

I don't understand your question so forgive me if my answer is
irrelevant.

Let

f[x_] = (2x - 5)/ (x - 2) ;

Then

Limit[f[x], x -> 3, Direction -> #] & /@ {-1, 1}
{1, 1}

since the function is continuus at x=3.

(Limit[f[x], x -> 2, Direction -> #1] & ) /@ {-1, 1}
{-Infinity, Infinity}

There is a pole of order one at x=2.

f[x] + O[x, 2]
SeriesData[x, 2, {-1, 2}, -1, 1, 1]

Normal[%]
2 - 1/(-2 + x)

Then from the last output isn't esy to figure out the behavior of the
function
as x approaching 2 from the left and right?

Consider also the following examples:

f2[x_] = Sin[x]/x
(Limit[f2[x], x -> 0, Direction -> #1] & ) /@ {-1, 1}
f2[x] + O[x, 0]
Normal[%]

Sin[x]/x
{1, 1}
SeriesData[x, 0, {1}, 0, 1, 1]
1

f3[x_] = Cos[x]/x
(Limit[f3[x], x -> 0, Direction -> #1] & ) /@ {-1, 1}
f3[x] + O[x, 0]
Normal[%]

Cos[x]/x
{Infinity, -Infinity}
SeriesData[x, 0, {1}, -1, 1, 1]
1/x

f4[x_] = Tan[x]/(x - Pi/2)
(Limit[f4[x], x -> Pi/2, Direction -> #1] & ) /@ {-1, 1}
f4[x] + O[x, Pi/2]
Normal[%]

Tan[x]/(-(Pi/2) + x)
{-Infinity, -Infinity}
SeriesData[x, Pi/2, {-1, 0, 1/3}, -2, 1, 1]
1/3 - 1/(-(Pi/2) + x)^2

In each Normal[f[x]+O[x,x0]] gives you the behavior of the function as
x approaching the singularity x0.
The first is a removable singulatity, the second a Cauchy-type
singularity and the third a double pole.

Note also

f5[x_] = Exp[-x]/x
Limit[f5[x], x -> Infinity]
f5[x] + O[x, Infinity]

1/(E^x*x)
0
Series::esss: Essential singularity encountered in E^SeriesData[x,
Infinity, \
{-1}, -1, 2, 1].
Series::esss: Essential singularity encountered in E^SeriesData[x, 0,
{-1}, \
-1, 2, 1].
Series::esss: Essential singularity encountered in E^SeriesData[x,
Infinity, \
{-1}, -1, 3, 1].
General::stop: Further output of Series::esss will be suppressed
during this \
calculation.
1/(E^x*x) + SeriesData[x, Infinity, {}, 1, 1, 1]


Best Regards
Dimitris




=CF/=C7 Mr Ajit Sen =DD=E3=F1=E1=F8=E5:
> Dear Sebastian,
>
>   Here is an example to illustrate what I meant:
>
>   f=(2x-5)/ (x-2)
>
>   Limit[f,x -> 3]  ---> 1
>                              Both agree here.
>   Normal[f+O[x,3]] ---> 1
>
>   However, at the point of discontinuity x = 2 (which
> I
>   referred to loosely as a "pole" : I find the whole
>   thing redolent of Laurent Series),
>
>   Limit[f,x -> 2]  ---> - Infinity   [Correct]
>
>   Normal[f+ O[x,2]]  ---> 2 - 1/(-2+x) [ = f ]
>
>   Now, I've always been using Normal to get rid of the
>   O[ ] terms in Series, and I found Andrzej's
>   alternative use of Normal rather neat, although I
>   have no idea how it works. Trace doesn't help me
> much
>   here.  So, the question is whether Normal can be
> used
>   at a point of discontinuity.
>
>   BTW, my query was prompted by Eric Smith's posts on
>   Limit.
>
>   Best Regards.
>
>   Ajit Sen.
>
>
>
>
>
>
>
>
> ___________________________________________________________
> Win a BlackBerry device from O2 with Yahoo!. Enter now. http://www.yahoo.=
co.uk/blackberry



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