       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:= Limit[Fibonacci[n+1]/Fibonacci[n],n->Infinity]

Fibonacci[1 + n]
Out= Limit[----------------, n -> \[Infinity]]
Fibonacci[n]

In:= Needs["Calculus`Limit`"]

In:= Limit[Fibonacci[n+1]/Fibonacci[n],n->Infinity]

Out= 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:= f[a_, b_, n_] := a^n + b^n

In:= lim[a_,b_]:=Module[{n},Limit[f[a,b,n+1]/f[a,b,n],n->Infinity]]

In:= mylim[a_,b_]/;Abs[a]>Abs[b]:=a

In:= mylim[a_,b_]/;Abs[a]<Abs[b]:=b

In:= mylim[a_,a_]/;a!=0:=a

In:= mylim[a_,b_]/;Abs[a]==Abs[b]:=Indeterminate

In:= comp[args__]:={{args},lim[args],mylim[args]}

In:= TableForm[Apply[comp, {{2, 3}, {(1 + Sqrt)/2, (1 -
Sqrt)/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//TableForm=
{2, 3}                       9                 3

1 + Sqrt  1 - Sqrt    3 (1 + Sqrt)   1 + Sqrt
{-----------, -----------}   ---------------   -----------
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>

```

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