Re: A limit bug

• To: mathgroup at smc.vnet.net
• Subject: [mg84526] Re: A limit bug
• From: Norbert Marxer <marxer at mec.li>
• Date: Wed, 2 Jan 2008 01:18:58 -0500 (EST)
• References: <flc7tk\$7vo\$1@smc.vnet.net>

```On 1 Jan., 03:18, "David W.Cantrell" <DWCantr... at sigmaxi.net> wrote:
> A recent question in sci.math led to something which should also interest
> this group.
>
> The OP asked about the limit of (p + q)!/(p! q!) as both p and q increase
> without bound. And he said later
>
> > I wasn't sure about it because Mathematica gives me a limit of zero.
> > Isn't that strange?
>
> I responded as follows.
>
> ----------------------------------------------
>
> Well, it's a bug. I suppose that what you did in Mathematica was something
> like
>
> In[3]:= Limit[Limit[(p + q)!/(p! q!), q -> Infinity], p -> Infinity]
>
> Out[3]= 0
>
> But note that there is not any way -- well, at least none known to me -- in
> Mathematica to get a true general "two-variable" limit:
>
> limit f(x,y) as (x,y) -> (x0,y0)
>
> However, Mathematica can get a correct answer for your limit problem.
>
> First, realize that (p + q)!/(p! q!) is Multinomial[p, q]. So you might try
>
> In[5]:= Limit[Limit[Multinomial[p, q], q -> Infinity], p -> Infinity]
>
> Out[5]= Limit[Limit[Multinomial[p, q], q -> Infinity], p -> Infinity]
>
> Since that remains unevaluated (but at least there was now no bug!), you
> might consider the possibility that it remained unevaluated for a good
>
> In[6]:= Limit[Limit[Multinomial[p, q], q -> Infinity, Assumptions -> p > 1],
> p -> Infinity]
>
> Out[6]= Infinity
>
> Success! Happy New Year!
>
> But BTW, note that, curiously, the following fails:
>
> In[7]:= Limit[Limit[(p + q)!/(p! q!), q -> Infinity, Assumptions -> p > 1],
> p -> Infinity]
>
> Out[7]= Indeterminate
>
> David

Hello

And what about telling Mathematica how p and q should approach
Infinity?

E.g. p and q approach Infinity in the same way:

Limit[(p + q)!/(p!*q!) /. q -> p, p -> Infinity]

E.g. q approaches Infinity faster than p:

Limit[(p + q)!/(p!*q!) /. q -> p^5, p -> Infinity]

... and many other ways would give the correct result, i.e. Infinity.

Best Regards
Norbert Marxer

```

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