Well, nobody has ever said that 1/0 will get you infinity, because it won't. Interpreted directly, it's an illegal operation, and there is no result because the computation is impossible. Sure enough, the gamma function doesn't exist at nonpositive integers.
But wait! If you interpret 1/0 in a limit sense... there is no result, because there's no constraint that the neighborhood of 0 is entirely positive or entirely negative (obviously, the neighborhood of 1 is entirely positive). You'd only get an answer of "infinity" if you were limiting a function like "1 / 0^2".
Sure enough, we see that the gamma function has no limit at the nonpositive integers, always approaching positive infinity from one side and negative infinity from the other side. So the factorial of a negative integer doesn't exist. What point are you trying to make?
> Sure enough, we see that the gamma function has no limit at
> the nonpositive integers, always approaching positive
> infinity from one side and negative infinity from the other side.
On the complex plane, there is more than one side to approach from. And the convention that was used when I learned complex analysis (stereographic projection), is that there is only one value of infinity, which can be approached from many directions. Under this model, the point at infinity is much better-behaved than the points at infinity in real analysis.
The Gamma function is characterized as having poles at the negative integers. That's a very straightforward description, and the behavior is well-understood and relatively easy to deal with. Certainly easier to deal with than the value of e^-x as x->0 (that singularity is essential, rather than a pole).
Yep, it sure does. The magnitude of Gamma(x) grows without bound as x approaches any negative real integer. The negative integers (and the point at infinity) are also the only points where Gamma(x) has a singularity.
