# Is this expression valid? [closed]

I encountered this expression in twitter,but I am not sure if it is correct.

$$\lim\limits_{x \to \infty} \frac{x!}{x^x}$$

The thing is, the factorial is only defined for positive integer numbers. I know that the gamma function is a generalizatión of the factorial, but in the expression of the image, it is not the gamma function, but the factorial that is being used.

Also, if we are to asume that this is an expression only for positive integer numbers, could we use the concept of "limit"? I think that this concept is reserved only to be used with real o complex numbers, is this correct?

Thank you everyone.

• Welcome to MSE. It is in your best interest that you type your posts (using MathJax) instead of posting links to pictures. Commented Jun 29 at 9:15
• Haven't you ever heard about limits of sequences? Commented Jun 29 at 9:17
• You can talk about limits of sequences - functions on the natural numbers. Perhaps $x \in \mathbb{N}$ here, which is sloppy variable use. Commented Jun 29 at 9:17
• After reading your question, Elon Musk decided that X was a more appropriate name for Twitter... Commented Jun 30 at 7:23

1. Who made the twitter post intended $$\lim_{x\to\infty}\frac{\Gamma(x+1)}{x^x}$$, and wrote $$x!$$ in a somewhat "captivating" way.
2. Who made the post intended a discrete limit, meaning $$\lim_{n\to\infty}\frac{n!}{n^n}$$, where $$n$$ is a natural number.