Answer
$0$
Work Step by Step
We are given the sequence:
$\left\{\dfrac{n^{1000}}{2^n}\right\}$
Let $a_n=n^{1000}$ and $b_n=2^n$.
As $n^{1000} \ll 2^n$, we have: $a_n\ll b_n$. Because $a_n$ appears before $b_n$ in the list of growth rates, using Theorem 8.6, we get:
$\lim\limits_{n \to \infty} \dfrac{a_n}{b_n}=0$
