Pedagogically speaking, I see two problems with defining
$\Gamma(z)$ (at least for real $z$) by the limit
$$\Gamma(z)=\lim_{m\to\infty}\frac{m! m^z}{\prod_{i=0}^m (z+i)}$$
as compared with the formula
$$\Gamma(z)=\lim_{m\to\infty}\frac{\lfloor m+z\rfloor! (z+m+1)^{\{z\}}}{\prod_{i=0}^m (z+i)}$$
(which one easily sees gives equivalent values in the limit). (The exponent
$\{z\}$ means the fractional part of $z$.) Of the two, only the
second formula yields $\Gamma(z+1)=z!$ for non-negative integers
$z$ via trivial limits of *constant* sequences. And so far as
I can see only the second formula arises immediately from a simple
one-sentence story that a student might use to discover the
formula as an exercise: march $\Gamma(z)$ to $\Gamma(z+m)$ using
$\Gamma(z+1)=z\Gamma(z)$, then estimate $\Gamma(z+m)$ as a
weighted geometric mean of the nearest factorials.

Questions: does the traditional limit support an equally compelling narrative? Ought one prefer it on other grounds? Neither limit serves well for numerical computation, but curiously, for a given $m$, they give errors of approximately equal magnitude but opposite sign. Does this have a conceptual explanation?