If $\left[x\right]$ is the greatest integer function, then $\underset{x\to 2^{+}}{\lim }\left(\frac{[x{]}^3}{3}-{\left[\frac{x}{3}\right]}^3\right)=$

Given $\left[x\right]$ is the greatest integer function

Then we have to find $\underset{x\to 2^{+}}{\lim }\left(\frac{[x{]}^3}{3}-{\left[\frac{x}{3}\right]}^3\right)$

Now $L=\underset{x\to 2^{+}}{\lim }\left(\frac{[x{]}^3}{3}-{\left[\frac{x}{3}\right]}^3\right)=\underset{h\to 0}{\lim }\left(\frac{[2+h{]}^3}{3}-{\left[\frac{2+h}{3}\right]}^3\right)$

Here $2<2+h<3$

$\Rightarrow \frac{2}{3}<\frac{2+h}{3}<1$

$\Rightarrow \left[\frac{2+h}{3}\right]=0$

and $2<2+h<3\Rightarrow \left[2+h\right]=2$

So $L=\frac{8}{3}$