Question

The set of all values of $a$ for which $\underset{x\to a}{\lim }([x-5]-[2x+2])=0$, where $[\propto ]$ denotes the greatest integer less than or equal to $\propto$ is equal to

• A. $[-7.5,-6.5)$
• B. $(-7.5,-6.5)$
• C. $(-7.5,-6.5]$
• D. $[-7.5,-6.5]$

Answer 1

Answer: (B)

$\underset{x\to a}{\lim }([x-5]-[2x+2])=0$
$\underset{x\to a}{\lim }([x]-5-[2x]-2)=0$
$\underset{x\to a}{\lim }([x]-[2x])=7$
$[a]-[2a]=7$
$a\in I,a=-7$
$a\notin I,a=I+f$
Now, $[a]-[2a]=7$
$-I-[2f]=7$
Case-I: $f\in \left(0,\frac{1}{2}\right)$
$2f\in (0,1)$
$-I=7$
$I=-7\Rightarrow a\in (-7,-6.5)$
Case-II: $f\in \left(\frac{1}{2},1\right)$
$2f\in (1,2)$
$-I-1=7$
$I=-8\Rightarrow a\in (-7.5,-7)$
Hence, $a\in (-7.5,-6.5)$