the trick is to "cancel the n's" so you have constant terms and things like: (something)/(a power of n) <-- all these types of terms go to 0 as n gets very large, so only the constant terms survive in the limit.
so (30/n)n = 30, -(8/n^2)(n(n+1)/2) = -(4n^2 + 4n)/n^2 = -4 - 4/n <--- the 4/n term will disappear in the limit.
and -(8/n^3)(n(n+1)(2n+1)/6) = -(8n^3 + 12n^2 + 4n)/(3n^3) = -8/3 - 4/n - 4/(3n^2) <--- the last two terms will disappear in the limit.
this leaves us with: 30 - 4 - 8/3 = (90 - 12 - 8)/3 = 70/3.
(if you carefully add the above, you get 70/3 - 8/n - 4/(3n^2), as your text indicates).