Find the limit of $(1+a_n)^{c_n}e^{-a_nc_n}$

Question

Let $a_n$ and $c_n$ be sequences of real numbers such that $a_n$ converges to $0$, and $c_na_n^2$ converges to $0$. Determine $\lim_{n\to\infty}(1+a_n)^{c_n}e^{-a_nc_n}$.

Answer 1

One may use a Taylor series expansion, as $n \to \infty$, $$ \begin{align} (1+a_n)^{c_n}e^{-a_nc_n}&=e^{c_n\log(1+a_n)}e^{-a_nc_n} \\&=e^{c_n(a_n+O(a^2_n))}e^{-a_nc_n} \\&=e^{c_na_n+O(c_na^2_n))}e^{-a_nc_n} \\&=e^{O(c_na^2_n)} \\&\to 1 \end{align} $$ since $a_n \to 0$ and $c_na_n^2 \to 0$.