<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] It is happens that there is an attractive closed form for $1^3+2^3+\cdots+n^3$, which can readily be proved by induction:
$$1^3+2^3+3^3+\cdots +n^3=\left(\frac{n(n+1)}{2}\right)^2.$$
Divide by $n^4$. Fairly quickly we find that the desired limit is $\dfrac{1}{4}$. 
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