A limit bug

*To*: mathgroup at smc.vnet.net*Subject*: [mg84511] A limit bug*From*: "David W.Cantrell" <DWCantrell at sigmaxi.net>*Date*: Mon, 31 Dec 2007 21:16:27 -0500 (EST)

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 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 David