Exploring the Limit as x Approaches Infinity of x!^(1/x) / x

The limit as x approaches infinity of the expression x!^(1/x) / x is an illuminating problem for understanding the behavior of factorials and their representation through Stirling's approximation. In this article, we will delve into the steps to find this limit, exploring various mathematical tools and approximation techniques to arrive at the solution.

Stirling's Approximation and Its Application

Stirling's approximation is a powerful tool in mathematics that provides an approximation for factorials of large numbers. It is given by:

n! ≈ √(2πn) (n/e)n

Using this approximation for x!, we have:

x! ≈ √(2πx) (x/e)x

To find the limit as x approaches infinity of the expression x!^(1/x) / x, we begin by taking the x-th root of x!. This step simplifies the factorial term significantly:

x!^(1/x) ≈ (sqrt(2πx) (x/e)x)1/x

x!^(1/x) ≈ (2πx)1/(2x) * (x/e)

Breaking this down further, we get:

(2πx)1/(2x) 2π1/(2x) * x1/(2x)

As x approaches infinity, both 2π1/(2x) and x1/(2x) approach 1. Therefore, we have:

x!^(1/x) ≈ x/e

Substituting this back into our original limit gives:

limx→∞ (x!^(1/x) / x) limx→∞ (x/e) / x 1/e

This leads us to the final answer:

limx→∞ (x!^(1/x) / x) 1/e

Alternative Method: Using Logarithms

Alternatively, one could approach this problem by taking the logarithm of the expression. Let:

f(x) (x!^(1/x) / x)

Then:

log(f(x)) (1/x) log(x!) - 1

Using Stirling's approximation for log(x!), we have:

log(x!) ≈ log(√(2πx) (x/e)x)

log(x!) ≈ (1/2) log(2πx) x log(x) - x log(e)

As x approaches infinity, the term (1/x) (1/2) log(2πx) approaches 0, and the term (1/x) x log(x) - x approaches -1. Therefore, we find that:

limx→∞ log(f(x)) -1

Thus:

limx→∞ f(x) e^(-1) 1/e

A Curious Definition and Its Limit

For variety and for fun, let's define:

mathcal{P}(m, n) Big[ prod_{k0}^{m-1} Big(1 - frac{k}{n}Big) Big]^{n/m^2}

This leads to the following limits:

limn→∞ mathcal{P}(n, n) 1/e

limm→∞ limn→∞ mathcal{P}(m, n) e^(-1/2)

This definition provides an interesting approach to understanding the behavior of the limit, showcasing the versatility and charm of mathematical exploration.