According to http://en.wikipedia.org/wiki/Binomial_distribution and http://mathworld.wolfram.com/BinomialDistribution.html , it seems like the following should evaluate to True:
Limit[FunctionExpand@PDF[BinomialDistribution[n, p], x],
n -> Infinity] ==
PDF[NormalDistribution[n p, Sqrt[n p (1 - p)]],
x], (n | x) \[Element] Integers && n >= 0 && 0 <= p <= 1 &&
p \[Element] Reals && x <= n && x >= 0]
I added all the assumptions I can think of, but it still doesn't evaluate. Am I making a mistake, or is Mathematica missing this identity?