Question

$\underset{x\to 2^{+}}{lim}(\frac{{\left[x\right]}^3}{3}-{\left[\frac{x}{3}\right]}^3)=?;$ ($\left[x\right]$ denotes greatest integer function)

• A. $0$
• B. $\frac{64}{27}$
• C. $\frac{8}{3}$
• D. $\frac{10}{3}$

Answer 1

Answer: (C)

$\frac{8}{3}$

$\underset{x\to 2^{+}}{lim}(\frac{{\left[x\right]}^3}{3}-{\left[\frac{x}{3}\right]}^3)$
$=\underset{x\to 2^{+}}{lim}\frac{{\left[x\right]}^3}{3}-\underset{x\to 2^{+}}{lim}{\left[\frac{x}{3}\right]}^3$
$=\frac{[2^{+}{]}^3}{3}-{\left[\frac{2^{+}}{3}\right]}^3=\frac{2^3}{3}-0=\frac{8}{3}$