Quotient with third order polynomial infinity as denominator confusion.

Question

The question concerned is:

$\lim\limits_{n \to \infty}[(\frac{1+\frac{1}{n}}{e})^n]^n$

I am reading about limits and have used l'hopitals rule to silmplify the original question to this:

$-\lim\limits_{n \to \infty}\frac{n^3}{2n^3+4n^2+2n}$

the solution says that this simplifies to $-\frac{1}{2}$

I don't understand how?

Thanks in advance.