Let $p(n,m)$ be the number of partitions of an integer $n$
into integers $\le m$, we have a well-known asymptotic expression: 

For a fixed $m$ and $n\to\infty$, 
$$p(n,m)=\frac{n^{m-1}}{m!(m-1)!} (1+O(1/n)) $$
 
My question is:  why the error $O(1/n)$ is independent of $m$?
Or how can it be extended for $m$ growing slowly with $n$?
Please help me to find the answer or the references. Thanks.