${\lim }_{n\to \infty }\left[\frac{1^3+2^3+3^3+\cdots +n^3}{n^4}\right]-{\lim }_{n\to \infty }\left[\frac{1^4+2^4+3^4+\cdots +n^4}{n^5}\right]$ is:

Express as Riemann sums (or use the power sum limit $\lim \frac{\sum r^k}{n^{k+1}}=\frac{1}{k+1}$):First limit: ${\lim }_{n\to \infty }\frac{1}{n}{\sum }_{r=1}^n(\frac{r}{n}{)}^3={\int }_0^1{x}^{3}dx=\frac{1}{4}$.Second limit: ${\lim }_{n\to \infty }\frac{1}{n}{\sum }_{r=1}^n(\frac{r}{n}{)}^4={\int }_0^1{x}^{4}dx=\frac{1}{5}$.Difference $=\frac{1}{4}-\frac{1}{5}=\frac{5-4}{20}=\frac{1}{20}$.