Re: Re: Limit question

*To*: mathgroup at smc.vnet.net*Subject*: [mg31603] Re: [mg31595] Re: Limit question*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Fri, 16 Nov 2001 06:38:08 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

This is a good explanation and much in the pragmatic spirit (the best output is the most useful one) which I find a rather attractive feature of Mathematica. But I am still a bit worried about one thing. You say that when the user specifies no direction in Limit, Mathematica automatically Defaults to Direction -> 1, and that the reason for this is that the other approach (take the user at his word and assume no direction) is generally less likely to produce useful answers and in those cases when it might the user can more profitably turn to Series. I completely agree. But of course the user can just as easily provide the direction if he really means to compute a directional limit. What's more, the fact that he should do is clearly documented, while the fact that he will get a directional limit even if he does not enter a direction does not seem to be. As far as I can see the only (useful?) result of this approach is to make this list a little more active. Andrzej Kozlowski On Thursday, November 15, 2001, at 07:52 PM, Daniel Lichtblau wrote: > > Some discussion has come up recently regarding the default behavior of > Limit. I thought I would say a bit about what it does, and why. > > First, as some people have noted, Limit[expr, x->xo] with no direction > specified behaves the same as Limit[expr, x->xo, Direction->-1] provided > x0 is finite. For x0=DirectedInfinity[foo] it approaches along a linear > path in the complex plane in the direction of foo (this may also have > been noted, I do not recall). > > Why must a direction be used? Well, for functions that are explicitly > not complex analytic (say, functions that use Abs or Sign) one can not > do much other than take a directional limit. I suppose one could take > two limits on the real line and check whether they agree. But a user can > do that just as well as we might. Let's move on to what I think is a > more important class for purposes of finding limits, complex analytic > functions. > > In this case there is little point to nondirectional limits. Either a > singularity is removable or there is no limit. Well, it's been pointed > out that the usual 1-point compactification with ComplexInfinity would > serve for poles. But Series can tell you well enough that there is a > pole, a result of ComplexInfinity is typically less useful than a > directed result, and this approach is utterly useless when confronted > with essential singularities. Yet this last class is quite important in > all sorts of asymptotic analysis. > > So what we do is akin to extending the complex numbers with a circle at > infinity which is a bit uncommon but seems to be the most reasonable way > to extend limit to get something useful. > > Another problem with NOT insisting on a path limit is that it is not > possible to know, for an expression such as Limit[1/x^2, x->0] whether x > is to be regarded as strictly real valued. Hence DirectedInfinity[1] > and DirectedInfinity[] (ComplexInfinity) might both be seen as viable > results. > > I will also point out that path dependence, or more generally, region > dependence of some sort, is required when one goes to multivariate > limits. In the common case one has meromorphic functions and is often > interested in behavior near points of indeterminacy (quotients of > vanishing holomorphic functions). While limits along paths are not quite > the most general case where one might obtain useful results, they are > all the same fairly useful ones for which limiting behavior will > actually exist. And again Series may be used to handle cases where > limits are trivial because there is no singularity. As in the univariate > case, ordinary poles will also not give useful information beyond an > undirected infinity unless one insists on a path-dependent result; > again, if a path independent infinity is desired, Series can be used to > make that determination. > > Whether or not people agree with the various explanations offered above > for why Limit does what it does, I hope this at least serves to remove > some of the mystery surrounding that default behavior. > > > Daniel Lichtblau > Wolfram Research > > > Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/