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