Question

The value of $\underset{x\to a}{lim}(\frac{|x{|}^3}{a}-{\left[\frac{x}{a}\right]}^3),a>0$
(where $[.]$ denotes the greatest integer function) is

• A. $a^2-3$
• B. $a^2-1$
• C. $a^2$
• D. does not exist.

Answer 1

The correct option is D does not exist.

Given : $\underset{x\to a}{lim}(\frac{|x{|}^3}{a}-{\left[\frac{x}{a}\right]}^3),a>0$

$R.H.L.=\underset{x\to a^{+}}{lim}\{\frac{|x{|}^3}{a}-{\left[\frac{x}{a}\right]}^3\}$
$=\underset{h\to 0}{lim}\{\frac{|a+h{|}^3}{a}-{\left[\frac{a+h}{a}\right]}^3\}$
$=\frac{a^3}{a}-1=a^2-1$
$L.H.L.=\underset{x\to a^{-}}{lim}\{\frac{|x{|}^3}{a}-{\left[\frac{x}{a}\right]}^3\}$
$=\underset{h\to 0}{lim}\{\frac{|a-h{|}^3}{a}-{\left[\frac{a-h}{a}\right]}^3\}$
$=\frac{a^3}{a}-0=a^2$

Clearly, $L.H.L.\ne R.H.L.$
$\therefore$ Limit does not exist.