Re: Got a trouble with the Limit[]
- To: mathgroup at smc.vnet.net
- Subject: [mg22808] Re: [mg22760] Got a trouble with the Limit[]
- From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
- Date: Fri, 31 Mar 2000 01:01:12 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
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. 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/