Calculus`Limit` is buggy!
- To: mathgroup at smc.vnet.net
- Subject: [mg13185] Calculus`Limit` is buggy!
- From: "Paul A. Rombouts" <paromb at worldonline.nl>
- Date: Mon, 13 Jul 1998 07:42:34 -0400
- Organization: World Online
- Sender: owner-wri-mathgroup at wolfram.com
In a message dated 7/4/98 8:01:56 PM, tobi.kamke at t-online.de wrote:
>I've a problem. I thought that Limit[Fibonacci[n]/Fibonacci[n-1], n ->
>Infinity] is GoldenRatio.
>
>Mathematica says 1.
>What's wrong?
Hi,
I use Mathematica 3.01 under Windows NT 4.0. I've tried to reproduce
your example, but I found out I needed to load the standard package
"Calculus`Limit`".
In[1]:= Limit[Fibonacci[n+1]/Fibonacci[n],n->Infinity]
Fibonacci[1 + n]
Out[1]= Limit[----------------, n -> \[Infinity]]
Fibonacci[n]
In[2]:= Needs["Calculus`Limit`"]
In[3]:= Limit[Fibonacci[n+1]/Fibonacci[n],n->Infinity]
Out[3]= 1
Well, I certainly found that last answer rather unsettling. I started
experimenting with some other limits and I got some very disturbing
results. I've summarised the results as follows:
In[4]:= f[a_, b_, n_] := a^n + b^n
In[5]:= lim[a_,b_]:=Module[{n},Limit[f[a,b,n+1]/f[a,b,n],n->Infinity]]
In[6]:= mylim[a_,b_]/;Abs[a]>Abs[b]:=a
In[7]:= mylim[a_,b_]/;Abs[a]<Abs[b]:=b
In[8]:= mylim[a_,a_]/;a!=0:=a
In[9]:= mylim[a_,b_]/;Abs[a]==Abs[b]:=Indeterminate
In[10]:= comp[args__]:={{args},lim[args],mylim[args]}
In[11]:= TableForm[Apply[comp, {{2, 3}, {(1 + Sqrt[5])/2, (1 -
Sqrt[5])/2},
{1/2, 1/3}, {5/6, 7/6}, {3, -5}, {-3, 5}, {I + 1, 2}, {2, 2},
{1/3, -1/3}, {1, I}, {5, 5*I}}, {1}], TableDepth->2]
Out[11]//TableForm=
{2, 3} 9 3
1 + Sqrt[5] 1 - Sqrt[5] 3 (1 + Sqrt[5]) 1 + Sqrt[5]
{-----------, -----------} --------------- -----------
2 2 2 2
1 1 1
{-, -} -
2 3 0 2
5 7 7 7
{-, -} - -
6 6 6 6
{3, -5} 0 -5
{-3, 5} 15 5
{1 + I, 2} ComplexInfinity 2
{2, 2} 2 2
1 1
{-, -(-)}
3 3 0 Indeterminate
{1, I} Indeterminate Indeterminate
{5, 5 I} 0 Indeterminate
In case it isn't clear from the from the preceding: mylim[a,b] gives the
value I think Limit[(a^(n+1)+b^(n+1))/(a^n+b^n),n->Infinity] should
have. Conclusion: If you use the package Calculus`Limit` be sure to
check your results with alternative methods.
greetings,
Paul A. Rombouts <P.A.Rombouts at phys.uu.nl>