Explanation

(d)
$\lim_{n \rightarrow \infty}{\frac{1 + 2^{4} + 3^{4} + ... + n^{4}}{n^{5}} - \lim_{n \rightarrow \infty}\frac{1 + 2^{3} + 3^{3} + ... + n^{3}}{n^{5}}}$
$\lim_{n \rightarrow \infty}{\frac{1}{n}\sum_{r = 1}^{n}{\left( \frac{r}{n} \right)^{4} -}}\lim_{n \rightarrow \infty}{\frac{1}{n} \times \lim_{n \rightarrow \infty}\frac{1}{n}\sum_{r = 1}^{n}\left( \frac{r}{n} \right)^{3}}$
$= \int_{0}^{1}{x^{4}dx -}\lim_{n \rightarrow \infty}\frac{1}{n} \times \int_{0}^{1}{x^{3}dx}$
$= \left\lbrack \frac{x^{5}}{5} \right\rbrack_{0}^{1} - 0 = \frac{1}{5}$