$\displaystyle \lim_{n\rightarrow \infty}\frac{n^{\frac{2}{3}}(\sqrt{n+2})}{2n^2+n+1 }$
How does the n^2/3 distribute?
Basically, if you take a positive expression under a square root, you double the exponent. In general:
$\displaystyle a^b=\sqrt[n]{a^{bn}}$ where $\displaystyle 0\le a^b$ if $\displaystyle n$ is even and $\displaystyle b$ is odd.