The following limit came up in the middle of solving a problem. Let $b$, $h$, and $k \geq 2$ be constants. If it helps simplifying the problem in any way, assume $b=2h$ and $k=2$.
I want to evaluate
$\underset{m \rightarrow +\infty}{\lim} \prod_{i=0}^m \left( \left(1-\frac{(1-2i)b}{4h}\frac{1}{m+1} \right)^{\left( \dfrac{4}{k^2}\dfrac{1}{m+1} - \dfrac{(1-2i)b}{hk^2}\dfrac{1}{(m+1)^2} \right)} \right).$
Numerical simulations show that this limit is a positive number less than 1. I would like to know how to evaluate this limit of product of infinite number of terms that approach 1.