Stirling's Formula (derived in Exercise 1.28), which gives an approximation for factorials, can be easily derived using the CLT.
(a) Argue that, if Xi ~ exponentiall),i = 1,2,..., all independent, then for every x,

where Z is a standard normal random variable.
(b) Show that differentiating both sides of the approximation in part (a) suggests

and that x = 0 gives Stirling's Formula

n-1 Vn T(n) e 2 /2