Find $\lim_{n\rightarrow \infty}\frac{an}{\log n}\left(1-\left(1-\frac{1}{an}\right)^{\log n}\right)$

Question

I would like to solve $$\lim_{n\rightarrow \infty}\frac{an}{\log n}\left(1-\left(1-\frac{1}{an}\right)^{\log n}\right)$$ to prove that $1-\left(1-\frac{1}{an}\right)^{\log n}$ is asymptotically equivalent to $\frac{\log n}{an}$. In fact Wolfram Alpha tells me that the limit is $1$, but I didn't manage to obtain it by pencil and paper.

Answer 1

Use Taylor expansions liberally (when it makes sense to): $$\left(1-\frac{1}{an}\right)^{\log n}=\exp\left[\log n\cdot\log\left(1-\frac{1}{an}\right)\right]\approx\exp\left[-\frac{\log n}{an}\right]\approx1-\frac{\log n}{an}$$