On a site generalizing the factorial function, it states, "The probability of repetition gets negligible; we find that the ratio between x!/(x−r)! and x^r approaches 1 as x grows large while r is held fixed." I was wondering how to formally prove this result.
Source: https://www.3blue1brown.com/sridhars-corner/2018/4/21/the-generalized-factorial
$$\begin{align}\lim_{x \to \infty} \frac{x!/(x-r)!}{x^r} &= \lim_{x \to \infty} \left[\frac{x}{x} \cdot \frac{x-1}{x} \cdot \frac{x-2}{x} \cdots \frac{x-r+1}{x}\right]\\ &= \lim_{x \to \infty} \frac{x}{x} \cdot \lim_{x \to \infty} \frac{x-1}{x} \cdot \lim_{x \to \infty} \frac{x-2}{x} \cdots \lim_{x \to \infty} \frac{x-r+1}{x} \end{align}$$