limit of a sequence when something about the limit is given Let $a_n$ be a sequence of real numbers such that
 $$\lim_{n\to\infty}|a_n+3((n-2)/n)^n|^{1/n}=\frac35. $$
Then what is $\lim_{n\to\infty}a_n$?

Call $b_n=((n-2)/n)^n$. You should know that $b_n \to e^{-2}$.
Now, you have for $n$ big enough
$$|a_n + b_n|^{1/n} < \frac 45$$
or equivalently,
$$|a_n + b_n| < \left( \frac 45 \right)^n \to 0$$
so by the squeeze theorem
$$\lim_n (a_n + b_n) = 0$$
Hence,
$$\lim_n a_n = - \lim_n b_n = -e^{-2}$$
