Question

$\underset{x\to a^{-}}{lim}\{\frac{|x{|}^3}{a}-{\left[\frac{x}{a}\right]}^3\},a>0$ is?

where [x] denotes the greatest integer less than or equal to $x$

• A. $a^2-3$
• B. $a^2-1$
• C. $a^2$
• D. None of the above

Answer 1

The correct option is C $a^2$

$\underset{x\to a^{-}}{lim}\{\frac{|x{|}^3}{a}-{\left[\frac{x}{a}\right]}^3\}$
${lim}_{h\to 0}\{\frac{|a-h{|}^3}{a}-{\left[\frac{a-h}{a}\right]}^3\}$
$={lim}_{h\to 0}\{\frac{a^3}{a}-{[1-\frac{h}{a}]}^3\}=a^2-{lim}_{h\to 0}0=a^2$