$\underset{n→a^-}{\lim}\left(\frac{|x|^3}{a}-\left[\frac{x}{a}\right]^3\right)(a > 0)$, where [*] denotes the greatest integer less than or equal to x, is

$a^2$

For $a – 1 < x < a,\left[\frac{x}{a}\right]=0$

$∴\underset{n→a^-}{\lim}\left(\frac{|x|^3}{a}-\left[\frac{x}{a}\right]^3\right)=\underset{n→a^-}{\lim}\left(\frac{|x|^3}{a}-0\right)$

$=\underset{h→0}{\lim}\left(\frac{|a-h|^3}{a}\right)=\frac{a^3}{a}=a^2$