My question comes from 1st expression in Page 227 of Durrett’s Probability textbook (5th edition).
It says: by taking $n \to \infty$ and $m/n \to x,$ we have $$\frac{n!}{m!(n-m)!} \frac{(m+1)!(n-m)!}{(n+2)!/2} \to 2x.$$ But if we do a reduction of LHS, we have $$\frac{m+1}{(n+2)(n+1)/2} = \frac{2(m/n + 1/n)}{(1 + 2/n)(n + 1)} \to \frac{2x}{\infty} = 0.$$
Am I wrong?
If not, this is an uncorrected typo since the 3rd edition.