Re: Got a trouble with the Limit[]
- To: mathgroup at smc.vnet.net
- Subject: [mg22864] Re: [mg22760] Got a trouble with the Limit[]
- From: "Kai G. Gauer" <gauer at sk.sympatico.ca>
- Date: Sun, 2 Apr 2000 15:33:34 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Andrzej Kozlowski wrote:
> on 00.3.24 5:28 PM, mason at mhl at tpts1.seed.net.tw wrote:
>
> > Dear all,
> > I had a trouble presenting the limit of a multiple defined function.
> > I set f[n_,x_]:=(1-(1+x)^-n)/x /; x>0&&x<1;
> > f[n_,x_]:=0 /; x<=0 || x>=1;
> > But I don't know how to show that
> > Limit[f[n_,x_], n->Infinity] equals to 1/x, when 0<x<1
> > Limit[f[n_,x_], n->Infinity] equals to 0, elsewhere.
> > Regards
> > Mason Lee
> >
> >
> >
>
> Are you sure this is right? (Your function is discontinuous at x=1).
> Anyway, what exactly do you mean by "I don't know how to show that..."? I
> assume that you know how to show this without using a computer just by
> referring to the elementary fact that a^(-x)->0 as x->Infinity if a>1 .
> Mathematica cannot tell you this since there is not way to pass to it the
> information that a>1 (Simplify with Assumptions in Mathematica 4 does not work with
> Limit[]). In addition there is a second problem, which is that
> Mathematica's Limit[] does not accept as input functions defined by multiple
> rules, even in cases like this:
>
> In[1]:= Clear[f]
>
> In[2]:= f[x_] /; x < 1 := x;
>
> In[3]:= f[x_] /; x >= 1 := x;
>
> In[4]:= Limit[f[x], x -> 1]
>
> Out[4]= Limit[f[x], x -> 1]
>
> One could easily teach Mathematica do deal with such trivial cases but in my
> opinion there is no point in doing so. In any case Mathematica cannot
> "really" find limits or "prove" much about them.
Everybody know that, but that's not really much of an excuse on
Mathematica's behalf! Remember the IEEE who brought us the standard
32-bit style of doing interval arithmetic the "best" way (their idea
was to choose the intervals most uniformily to get "nice"
approximations when searching for k points of a big "interval" of
2^2^(huh) (where huh is your favourite big number that would best
satisfy you)). Unfortunately, this seems to be partially
pseudo-excusable on Mathematica's behalf. After all, who really needs
to print off the first ten billion digits of pi to your nearest t-shirt
when you have the Ramanujan formula for getting the nth digit (much
easier, since it's not as recursively dependent)?!?!?!?!! If you only
need to plot or something, do a couple hundred sample points at most
and hope for the best (if you can't hope for the best, chances are that
you probably won't be able to prove your search results too much more
rigourously either...look at the latest search of lim tan(x)/x^8 in the
MAA periodicals). Most computer screens or printers (nor people who
build them) just won't build a stronger resolution monitor, simply
because we all ready have the best there is (limited to our eyes and
the size of our audio cards). If you're looking at proofs, use
L'hopital's rule a few hundred times (with approximations of limits
subbed in when you don't know the exact value, and of course, you
usually won't, unless you get very lucky in your differentiation). By
the way, L'Hospital's Rule doesn't always work well, or even any better
than when you started with: I know of at least one function (doesn't
involve dirichlet type functions or anything like that, I even suspect
it's quite continuous for most points, though I can't find out what it
is) ..something to the effect of now matter how you split it up into
numerator and denominator (where both are going to 0 or oo), there
aren't too many obvious ways to split it in such a way that the nth
application ofL'Hospital's derivative tells you exactly how much better
you may or may not be getting. What I do remember about this function,
however, is that after applin Hospital's rule twice (running the test
to make sure it's still of proper form) is that you get the same
function back as what you started with. To this day, I know of no way
to choose the numerator and denominator to get rid of this sort of
periodicity, to extend the interval (so that we may not notice this for
two or more iterations) or whether when finding one such num-denom form
means that we'll ALWAYS find other similar num-denom forms. Like I
said, I don't explicitly have the nice, relatively simple to state fcn
on me but I o know that it exists. However, that's basically all I know
about it. If you know other things about this bad behaviour, please
tell me!
> Limits properly speaking
> belong to analysis which deals with continuous phenomena while computers are
> by nature discrete. Mathematica can only deal with a relatively small number
> of cases which can be reduced to some basic facts that are a part of its
> "knowledge data base" by applying certain algebraic ("discrete")
> procedures.( One example of such an algebraic procedure is the "L'Hospital
> rule"). However, genuine proofs in analysis take the form of "epsilon-delta"
> arguments, only the simplest cases of which can at this time be tackled by
> "theorem proving" systems and none at all by Mathematica. In any case one
> should never try to use a computer to do something that is easy to do by
> hand, particularly in mathematics.
>
> --
> Andrzej Kozlowski
> Toyama International University
> Toyama, Japan
> http://sigma.tuins.ac.jp/