Find the limit of $\lim_n a_n$ 
Let , $\{a_n\}$ be a sequence of real numbers such that $$\lim_n \left|a_n+3\left(\frac{n-2}{n}\right)^n\right|^{1/n}=\frac{3}{5}.$$Compute $\displaystyle \lim_na_n$.

I have no idea about this..Please help.

Let $b_n = a_n + 3\left(\frac{n-2}{n}\right)^n = a_n + 3 \left(1-\frac{2}{n}\right)^n$.
We know that $b_n^{1/n} \xrightarrow[n\to\infty]{} \frac{3}{5}$, and therefore $b_n \sim_{n\to\infty} \left(\frac{3}{5}\right)^n \xrightarrow[n\to\infty]{} 0$. It follows that, since $3\left(\frac{n-2}{n}\right)^n \xrightarrow[n\to\infty]{}  3e^{-2}$ (why?), we must have $a_n \xrightarrow[n\to\infty]{} - 3e^{-2}$.
