# Thread: What is this limit from?

1. ## What is this limit from?

How is the limit = 2/3? Im not sure if l'hopital was used or how the 3^(n+1) affects the limit...

2. ## Re: What is this limit from?

first off realize that

$\lim \limits_{n\to \infty} \dfrac{n^k}{3^n} = 0,~\forall k$

next divide top and bottom by $3^n$ to obtain

$\lim \limits_{n \to \infty}~\dfrac{2\left(1-\frac{n^3}{3^n}\right)}{3 - \frac{n^3}{3^n} - \frac{3n^2}{3^n} - \frac{3n}{3^n} - \frac{1}{3^n}}= \dfrac 2 3$