n! Open n to the power (n→∞), how to find the limit?

The limit of the expression \( n! / n^n \) as \( n \to \infty \) is zero. This result is not immediately obvious from inspection, as both the numerator and denominator grow without bound, but it can be rigorously established by recognizing that the factorial in the numerator grows significantly slower than the exponential form \( n^n \) in the denominator. The factorial \( n! = n \times (n-1) \times \cdots \times 2 \times 1 \) is a product of \( n \) factors, each at most \( n \), whereas \( n^n \) is the product of \( n \) factors all exactly equal to \( n \). The key is that for large \( n \), most factors in \( n! \) are substantially smaller than \( n \), causing the ratio to decay rapidly toward zero.

A direct analytical approach involves rewriting the ratio to bound it. One can observe that \( n! / n^n = (n/n) \times ((n-1)/n) \times \cdots \times (1/n) \). This is a product of \( n \) terms, each of the form \( k/n \) for \( k = 1, 2, \dots, n \). For \( k \leq n/2 \), the terms are at most \( 1/2 \), and there are approximately \( n/2 \) such terms, suggesting an exponential decay. More formally, one can use the inequality \( n! \leq n^{n-1} \) for \( n \geq 2 \), which gives \( n! / n^n \leq 1/n \), directly implying the limit is zero by the squeeze theorem. A sharper bound can be obtained via Stirling's approximation, \( n! \sim \sqrt{2\pi n} (n/e)^n \), which provides an asymptotic formula: \( n! / n^n \sim \sqrt{2\pi n} \, e^{-n} \). This approximation makes the convergence to zero explicit, as the dominant term \( e^{-n} \) decays exponentially, overwhelming the algebraic growth of \( \sqrt{n} \).

The implications of this limit extend to probability and combinatorial analysis. For instance, it quantifies the vanishing probability that a random function from a set of \( n \) elements to itself is a permutation (a bijection), as there are \( n! \) permutations and \( n^n \) total functions. The result confirms the intuitive notion that for large sets, a randomly selected function is overwhelmingly likely not to be one-to-one. Mechanically, the convergence rate is extremely fast due to the exponential decay, meaning that even for moderately sized \( n \) (e.g., \( n=10 \)), the ratio is already on the order of \( 10^{-4} \), and for \( n=100 \), it is astronomically small.

In summary, the limit is definitively zero, established through product decomposition, simple inequalities, or asymptotic approximation. The analysis underscores a fundamental growth hierarchy: factorial growth, while super-exponential, is strictly dominated by the growth of \( n^n \), which itself is a form of exponential growth with a base increasing with \( n \). This hierarchy is crucial in algorithmic complexity and statistical mechanics, where such ratios often delineate feasible from infeasible configurations. The result is robust and verifiable through multiple mathematical lenses, each reinforcing the same conclusion without ambiguity.