Question

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. Limit does not exist

Answer 1

The correct option is D Limit does not exist

$\underset{x\to a}{lim}\{\frac{|x{|}^3}{a}-{\left[\frac{x}{a}\right]}^3\}$

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

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