# Proving this Limit

1. Sep 29, 2004

### relinquished™

I'm a little stuck in my proof here. As I was trying to prove that the limit of a binomial distribution is the poisson distribution, I encountered this:

$$\lim_{n\to +\infty} \frac{n!}{(n-x)! (n-k)^x}$$

where x and k are arbitrary constants.

The books say that this approaches 1, but shows no formal proof. How are we sure that this approaches 1 as n gets larger? In short, what's the formal proof?

Thanx for any help

2. Sep 29, 2004

### Hurkyl

Staff Emeritus
Let's see... you have n things multiplied together on the top (1, 2, 3, ..., n), and you have n things multipled together on the bottom: (n - x) of them in (n-x)! and x of them in (n-k)^x. My first instinct would be to try and group terms in the numerator with terms in the denominator.