Question

$\underset{x\to 2^{+}}{lim}(\frac{|x{|}^3}{3}-{\left[\frac{x}{3}\right]}^3),$ where $\left[x\right]$ is the greatest integer less than or eqaul to $x$ is :

• A. 5/3
• B. 8/3
• C. 7/9
• D. None of these

Answer 1

The correct option is B 8/3

$\underset{x\to 2^{+}}{lim}(\frac{|x{|}^3}{3}-{\left[\frac{x}{3}\right]}^3)$
Now for $x⟶2^{+}x>0\Rightarrow |x|=x$
and $0<\left[\frac{x}{3}\right]<1\Rightarrow \left[\frac{x}{3}\right]=0$
$\Rightarrow \underset{x\to 2^{+}}{lim}(\frac{|x{|}^3}{3}-{\left[\frac{x}{3}\right]}^3)=\underset{x\to 2^{+}}{lim}\left(\frac{x^3}{3}\right)=\frac{8}{3}$