We define for all real $x > 0$ the sequences $(I_{n}(x))_{n \geqslant 1}$ and $(J_{n}(x))_{n \geqslant 0}$ by:
$$I_{n}(x) = \int_{0}^{n} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} dt, \qquad J_{n}(x) = \int_{0}^{1} (1-t)^{n} t^{x-1} dt$$
Establish Euler's identity:
$$\forall x > 0, \quad \Gamma(x) = \lim_{n \rightarrow +\infty} \frac{n! \, n^{x}}{x(x+1) \cdots (x+n)}$$