  The singularities of the $\Gamma$ function, a meromorphic function on the
complex plane, are known to occur at the nonpositive integers. We show, using
Euler and Gauss identities, that for all positive integers $n$ and $k$, $$
\lim_{z\rightarrow 0} \frac{\Gamma(nz)}{\Gamma(z)} = \frac 1 n; \hspace{0.4in}
\lim_{z\rightarrow -k} \frac{\Gamma(nz)}{\Gamma(z)} = \f{(-1)^{k}\
\Gamma(k)}{n^2\ \Gamma(nk)}.$$
  The above relations add to the list of the known fundamental Gamma function
identities.
