Re: Limit question

*To*: mathgroup at smc.vnet.net*Subject*: [mg31595] Re: Limit question*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Thu, 15 Nov 2001 05:52:28 -0500 (EST)*References*: <200111110534.AAA13842@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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

**References**:**Re: Limit question***From:*Andrzej Kozlowski <andrzej@tuins.ac.jp>