Evaluate $\lim\limits_{n\rightarrow \infty}\left\{\frac{1^3}{n^4}+\frac{2^3}{n^4}+\frac{3^3}{n^4}+\dots +\frac{n^3}{n^4}\right\}$ 
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*Evaluate $\displaystyle\lim_{n\rightarrow \infty}\left\{\frac{1^3}{n^4}+\frac{2^3}{n^4}+\frac{3^3}{n^4}+\dots +\frac{n^3}{n^4}\right\}$.

*Examine whether $x^{1/x}$ possesses a maximum or minimum and determine the same. 

This begs to be viewed as a Riemann sum:
$$
\frac 1n\left(\frac{1^3}{n^3}+\cdots+\frac{n^3}{n^3}\right) \to \int_0^1 x^3\,dx.
$$
Your second question seems quite different from your first and should be posted separately.
