The left hand limit at x = a for the function $f(x)=\left\{\begin{matrix}\frac{|x|^3}{a}-\left[\frac{x}{a}\right]^3\end{matrix}\right\}(a > 0)$, where [x] denotes the greatest integer less than or equal to x is

$a^2$

For left hand limit x < a i.e., x = a − h, where h → 0 and a > 0

∴ x is positive |x|= x

Also, $\frac{x}{a}<1$ but it is positive $\frac{x}{a}$ lies between 0 and 1 so that $\left[\frac{x}{a}\right]=0$

$∴\lim f(x)=\frac{a^3}{a}-0=a^2$