I am studying the proof of Lloyd and Shepp of the central limit theorem for number of cycles in random permutations. (paper can be found here https://www.ams.org/journals/tran/1966-121-02/S0002-9947-1966-0195117-8/S0002-9947-1966-0195117-8.pdf) if we define $\sigma$ as the total number of cycles, then
following the calculations, they get
$\mathbb{E}_N(\exp{(it \sigma)}) \approx \frac{N^{e^{it}-1}}{\Gamma(e^{it})}\left( 1 + \frac{\theta_N(t)}{N-1}\right)$
then, for the normalized random variable $\frac{\sigma- \log N}{\sqrt{\log N}}$ one should get
$\lim_{N \rightarrow \infty}\mathbb{E}_N \left( \exp\left(it \frac{\sigma- \log N}{\sqrt{\log N}} \right)\right) = \exp{\left(- \frac{1}{2}t^2 \right)}$.
Could someone explain how one can arrive at the last step?