How can I compute an asymptotic estimate for following expression?
\begin{equation} A = \frac{(1-\frac{1}{s})(1-\frac{2}{s})...(1-\frac{t-1}{s})}{(1-\frac{1}{n-s})(1-\frac{2}{n-s})...(1-\frac{t-1}{n-s}) } \end{equation}
We know that $n \gg 1, s \gg 1, t \ll n, t \ll s.$
Assuming $t\ll n-s$, $$ \begin{align} \log\left(\left[1-\tfrac1s\right]\left[1-\tfrac2s\right]\cdots\left[1-\tfrac{t-1}s\right]\right) &=-\left(\tfrac1s+\tfrac2s+\dots+\tfrac{t-1}s\right)+O\left(\frac{t^3}{s^2}\right)\\ &=-\frac{t(t-1)}{2s}+O\left(\frac{t^3}{s^2}\right) \end{align} $$ and $$ \begin{align} \log\left(\left[1-\tfrac1{n-s}\right]\left[1-\tfrac2{n-s}\right]\cdots\left[1-\tfrac{t-1}{n-s}\right]\right) &\sim-\left(\tfrac1{n-s}+\tfrac2{n-s}+\dots+\tfrac{t-1}{n-s}\right)+O\left(\frac{t^3}{(n-s)^2}\right)\\ &=-\frac{t(t-1)}{2(n-s)}+O\left(\frac{t^3}{(n-s)^2}\right) \end{align} $$ Thus, $$ \begin{align} &\log\left(\frac{\left[1-\tfrac1s\right]\left[1-\tfrac2s\right]\cdots\left[1-\tfrac{t-1}s\right]}{\left[1-\tfrac1{n-s}\right]\left[1-\tfrac2{n-s}\right]\cdots\left[1-\tfrac{t-1}{n-s}\right]}\right)\\[6pt] &=\frac{t(t-1)}2\frac{2s-n}{s(n-s)}+O\left(\frac{t^3}{s^2}\right)+O\left(\frac{t^3}{(n-s)^2}\right) \end{align} $$ and depending on how much smaller $t$ is than $n-s$ and $s$, for example, if $t=o(s^{2/3})$ and $t=o\left((n-s)^{2/3}\right)$, we would have $$ \frac{\left[1-\tfrac1s\right]\left[1-\tfrac2s\right]\cdots\left[1-\tfrac{t-1}s\right]}{\left[1-\tfrac1{n-s}\right]\left[1-\tfrac2{n-s}\right]\cdots\left[1-\tfrac{t-1}{n-s}\right]} \sim\exp\left(\frac{t(t-1)}2\frac{2s-n}{s(n-s)}\right) $$