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I am not sure what you mean by "there may be other

approaches". binomial(a,a) should simplify to Q if and only if

the double limit exists:

limit binomial(x,y) = Q

x->a

y->a

A necessary (but in general not sufficient) condition for this

to exist is that the two single limits exist and are equal:

limit binomial(a,y) = limit binomial(x,a) = Q

y->a x->a

If the limit is not well-defined, then binomial(x,x) is not well-

defined, and it will cause incorrect results in some case or

another to arbitrarily set it to some value.

I don't know if this limit is or isn't well-defined. I do know

that depending on identities with unspecified domains of

validity is dangerous....