<s>[INST] Proof: for a pretty nasty limit Let $$ f(x) = \lim_{n\to\infty} \dfrac {[(1x)^2]+[(2x)^2]+\ldots+[(nx)^2]} {n^3}$$.
Prove that f(x) is continuous function.
Edit: $[.] $ is the greatest integer function.
 [/INST] $k^2x^2-1 < [(kx)^2] \le k^2x^2$
Summing the three expressions from $k=1$ to $n$ and dividing by $n^3$, we get
$\frac{x^2}{6}(1+\frac{1}{n})(2+\frac{1}{n}) - \frac{1}{n^2}< \frac{[(1x)^2] + \cdots + [(nx)^2]}{n^3} \le \frac{x^2}{6}(1+\frac{1}{n})(2+\frac{1}{n}) $.
Taking limits as $n \rightarrow \infty$, we get
$\frac{x^2}{3} \le f(x) \le \frac{x^2}{3}$ for every $x \in \mathbb{R}$. Thus $f(x)=\frac{x^2}{3}$ which is a continuous function.
(Note that the above, also proves that the limit always exists.)
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