Exponential ( Problem with the mathematical description)

Question

Well, I know the mathematical description of e as $\lim_{n\to\infty}(1-\frac{m}{n})^{n}=e^{-m}$ ,but today in my statistical mechanics class, while calculating the volume of thin shell of thickness 's' of n-dimensional hypersphere of radius R, he used this expression $(1-\frac{s}{R})^n=e^{-\frac{sn}{R}}$ I haven't been able to figure it out on how did it come from, since it doesn't match with the definition of $e^{-m}$

Answer 1

$\lim\left(1-\frac{s}{R}\right)^n = \lim\left(\left(1-\frac{s}{R}\right)^R\right)^{\frac{n}{R}} = \left(e^{-s}\right)^{\frac{n}{R}} = e^{-\frac{sn}{R}}$