Why does $p$ have to be moderate in the Poisson approximation to binomial random variable? So the proof that a binomial rv with large $n$ approximates a poisson rv with $\lambda = np$ (given below) doesn't seem to use the fact that $p$ is moderate/small, so why does wikipedia and my textbook (Ross) state this as a condition?
Proof:
If $\lambda = np$, then $ P(X = i) = {n \choose i} p^i (1 - p)^{n-i} $
$$ = \frac{(n(n-1) ... (n  - i + 1)}{i!} (\frac{\lambda}{n})^i \frac{(1 - \frac{\lambda}{n})^n}{(1 - \frac{\lambda}{n})^i} $$
and since $ \lim_{n\to \infty} \frac{(n(n-1) ... (n  - i + 1)}{n^i} = 1$
and $ \lim_{n\to \infty} (1 - \frac{\lambda}{n})^i = 1$
$$ \lim_{n\to \infty} P(X = i) = \frac{\lambda^i}{i!}e^{-\lambda}$$

Suppose for example that $p=1/2$ and $n$ is say $100$. Let $i=50$. Then $\frac{n(n-1)(n-2)\cdots(n-i+1)}{n^i}$ is quite far from $1$.  So the argument quoted no longer works.
