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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
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