Physicist: Need help understanding a line in a derivation

Question

The derivation I'm looking at is the Poisson distribution in M. Fox's "Quantum Optics" Chapter 5.

Now by Stirling's Formula:

$\lim_{N \to \infty} [\ln N!] = N \ln N - N $

we can see that

$\displaystyle\lim_{N \to \infty} \big[\ln \big(\frac{N!}{N^n(N-n)!}\big)\big] = 0$

It's not clear to me why this limit equates to zero. Am I missing something obvious?

Answer 1

$$\begin{align}\log{\left (\frac{N!}{N^n (N-n)!} \right )} &= \log{N!} - n \log{N} - \log{(N-n)!} \\ &\approx N \log{N}-N - n \log{N} - (N-n) \log{(N-n)} + (N-n) \text {[By Stirling's Formula]}\\ &= (N-n) \log{N} - (N-n) \log{(N-n)} - n \\ &= (N-n) \log{\left (\frac{N}{N-n} \right )} - n \\ &= -(N-n) \log{\left (1-\frac{n}{N} \right )} - n \\ &\approx -N \left ( -\frac{n}{N}\right ) - n\end{align}$$

Thus the limit is indeed $0$ as $N \to \infty$.