Euler–Gauss limit formula

Remarks

For $\operatorname{Re} z > 0$, the quotient equals $\int_0^n t^{z-1} (1 - t/n)^n \, dt$, by repeated integration by parts. Since $(1 - t/n)^n$ increases to $e^{-t}$, the dominated convergence theorem gives the limit $\int_0^\infty t^{z-1} e^{-t}\,dt = \Gamma(z)$. For other $z$, the functional equation $\Gamma(z + 1) = z\Gamma(z)$ moves the argument into the half-plane, and the quotients transform in the same way up to a factor that tends to $1$. Example: at $z = 1$ the quotient is $\frac{n!\, n}{(n+1)!} = \frac{n}{n+1} \to 1 = \Gamma(1)$. In Mathlib the statement is Complex.GammaSeq_tendsto_Gamma.

Going deeper

For a student

The limit makes sense for every allowed complex $z$ at once, so it can serve as a definition of the meromorphic gamma function. Taking reciprocals and rewriting $n^z = \exp\big(z(\ln n - 1 - \tfrac12 - \cdots - \tfrac1n)\big)\exp\big(z(1 + \cdots + \tfrac1n)\big)$ turns it into the Weierstrass product. It is also the limit that appears in the proof of the Bohr–Mollerup theorem.