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A limit bug

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

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
reason, namely, because a little more information had to be provided:

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


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