<s>[INST] 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. 
 [/INST] $\displaystyle\lim_{n\rightarrow \infty}\left\{\frac{1^3}{n^4}+\frac{2^3}{n^4}+\frac{3^3}{n^4}+\dots +\frac{n^3}{n^4}\right\}$
= $\displaystyle\lim_{n\rightarrow \infty}\left\{\frac{1^3+2^3+3^3+\dots +n^3}{n^4}\right\}$  =
 $\displaystyle\lim_{n\rightarrow \infty}\left\{\left(\frac{n(n+1)}{2n^2}\right)^2\right\}$
=$\frac{1}{4}\displaystyle\lim_{n\rightarrow \infty}\left(1+\frac{1}{n^2} \right)^2$
=$\frac{1}{4}$
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