Verification of poisson approximation to hypergeometric distribution

How can I verify that

[tex]\lim_{N,M,K \to \infty, \frac{M}{N} \to 0, \frac{KM}{N} \to \lambda} \frac{\binom{M}{x}\binom{N-M}{K-x}}{\binom{N}{K}} = \frac{\lambda^x}{x!}e^{-\lambda}[\tex],

**without** using** Stirling's formula **or the **Poisson approximation to the Binomial**?

I have been stuck on this problem for a while, because I don't know how to divide up the terms and factorials without using the help of prior results!

Any help would be appreciated. Thanks in advance.

Re: Verification of poisson approximation to hypergeometric distribution

Hey abscissa.

I can't quite see the latex (it might just be my end), but one way of doing these kinds of problems is to look at the moment generating function.

In fact the Central Limit Theorem can (and is often) be proved by showing that the moment generating function in the limit approaches, equal, or is approximate to the MGF of the target distribution.