In the 9 lecures in random graphs on pages 16/17
http://www.iecn.u-nancy.fr/~chassain/GDT/documents/SpencerStFlour.pdf
they say let $n_{0}(k)$ be the minimum $n$ for which $\binom{n}{k} 2^{-\binom{k}{2}} \geq 1$. Then it says that for any $\lambda \in (-\infty,\infty)$ let $n=n_{0}(k)\left[1+\frac{\lambda+o(1)}{k}\right]$ it then follows that $\binom{n}{k}=\left[1+\frac{\lambda+o(1)}{k}\right]^{k}=e^{\lambda}+o(1).$
I am completely unsure as to how they arrived at the last part. Any ideas?
$$2^{-k(k-1)/2}\cdot{n\choose k}=2^{-k(k-1)/2}\cdot{n_0(k)\choose k}\cdot\frac{n!}{(n-k)!}\cdot\frac{(n_0(k)-k)!}{n_0(k)!}\sim1\cdot\left(\frac{n}{n_0(k)}\right)^k $$