Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$) I apoligize if this is a stupid/obvious question, but last night I was wondering how we can compute limits for factorials of negative integers, for instance, how do we evaluate:
$$\lim_{x\to-3}\frac{x!}{(2x)!}=-120$$
Neither $x!$, nor $(2x)!$ are defined for $x\in\mathbb{Z}^{-}$, and indeed, both are singularities according to the graph of $\Gamma(x+1)$. 
The book I am reading calculates this using a previously shown identity that:
$$F\left(\left.{1-c-2n,-2n \atop c}\right|-1\right)=(-1)^{n}\frac{(2n)!}{n!}\frac{(c-1)!}{(c+n-1)!},\space\forall n\in\mathbb{Z}^{*}$$
And then, the more general Kummer's Formula:
$$F\left(\left.{a,b \atop 1+b-a}\right|-1\right)=\frac{(b/2)!}{b!}(b-a)^{\underline{b/2}}$$
It then shows that they would only produce consistent results if:
$$(-1)^{n}\frac{(2n)!}{n!}=\lim_{b\to-2n}{\frac{(b/2)!}{b!}}=\lim_{x\to-n}{\frac{x!}{(2x)!}},\space n\in\mathbb{Z}^{*}$$
It then gives the example of $n=3$, proving that:
$$\lim_{x\to-3}{\frac{x!}{(2x)!}}=-\frac{6!}{3!}=-120$$
However, using Wolfram|Alpha, I can see that there are other such limits defined (such as $\lim_{x\to-3}{\frac{x!}{(8x)!}}=-103408066955539906560000$.
Without using the hypergeometric series, how could we evaluate limits such as these?
Again, sorry if this is a stupid question, thanks in advance!
 A: Using anon's "pole" idea, with definition $x! := \Gamma(x+1)$ we have:
$$\begin{align}
x! &= \Gamma(x+1) = \frac{1}{2}\;\frac{1}{x+3}+O(1)\qquad\text{as $x \to -3$},
\\
(2x)! &= \Gamma(2x+1) = -\frac{1}{240}\;\frac{1}{x+3} + O(1)\qquad\text{as $x \to -3$},
\\
\frac{x!}{(2x)!} &= \frac{1/2}{-1/240}+O(x+3)\qquad\text{as $x \to -3$},
\\
\frac{x!}{(2x)!} &\to -120\qquad\text{as $x \to -3$}.
\end{align}$$
A: You want to compute $\displaystyle \lim_{x\to -n} \frac {\Pi(x)}{\Pi(mx)}$ when $x$ is near a negative integer.
$\Pi$ is the 'natural' extension of the factorial : $\Pi(n)=n!$ and $\Pi(z)=\Gamma(z+1)$ (see Wikipedia)
In this form the "Euler's reflection formula" becomes simply (for $\operatorname{sinc}(z)=\frac{\sin(\pi z)}{\pi z}$) :
$$\Pi(-z)\Pi(z)=\frac 1{\operatorname{sinc}(z)}$$
$$ \lim_{x\to -n}\ \frac {\Pi(x)}{\Pi(mx)}=\lim_{x\to -n}\frac {\Pi(-mx)\operatorname{sinc}(-mx)}{\Pi(-x)\operatorname{sinc}(-x)}$$
 $$ =\lim_{t\to n}\frac {\Pi(mt)\operatorname{sinc}(mt)}{\Pi(t)\operatorname{sinc}(t)}$$
It remains to prove that $\ \lim_{t\to n} \frac {\operatorname{sinc(mt)}}{\operatorname{sinc(t)}}=\frac {(-1)^{(m-1)n}}m$ (you may use l'Hôpital's rule for that) and to conclude! 
A: Let me add to the above, that while $\Gamma(-n)$ for a positive integer $n$ is undefined, let $m$ be such an integer as well, and then the ratio $\Gamma(-n)/\Gamma(-m)$ is well defined, and the Euler reflection formula above leads to its value being equal to $\Gamma(m+1)/\Gamma(n+1)(-1)^{n-m}$.
     This shows, by the way that the ratio mentioned at the beginning of the this sequence, effectively $(-3)!/(-6)!$ is $-60$, and not as suggested.
A: I think that the limit formulas
\begin{equation*}%\label{gamma-limit-eq}
\lim_{z\to-k}\frac{\Gamma(nz)}{\Gamma(qz)}=(-1)^{(n-q)k}\frac{q}{n}\frac{(qk)!}{(nk)!}, \quad k\in\{0,1,2,\dotsc\} \quad n,q\in\{1,2,\dotsc\}
\end{equation*}
and
\begin{equation}\label{polygamma-limit-eq}
 \lim_{z\to-k}\frac{\psi(nz)}{\psi(qz)}=\frac{q}{n}, \quad k\in\{0,1,2,\dotsc\} \quad n,q\in\{1,2,\dotsc\}
\end{equation}
give a perfect answer. One can find alternative proofs of these limit formulas in the papers [1, 2, 3] below.
References

*

*A. Prabhu and H. M. Srivastava, Some limit formulas for the Gamma and Psi (or Digamma) functions at their singularities, Integral Transforms Spec. Funct. 22 (2011), no. 8, 587--592; available online at https://doi.org/10.1080/10652469.2010.535970.

*F. Qi, Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities, Filomat 27 (2013), no. 4, 601--604; available online at http://dx.doi.org/10.2298/FIL1304601Q.

*L. Yin and L.-G. Huang, Limit formulas related to the $p$-gamma and $p$-polygamma functions at their singularities, Filomat 29 (2015), no. 7, 1501--1505; available online at https://doi.org/10.2298/FIL1507501Y.

