Limits Question 2
Question 2 - 24 January - Shift 2
The set of all values of a for which
$lim_{x \to a}([x-5]-[2 x+2])=0$, where $[a]$ denotes the greater integer less than or equal to $a$ is equal to
(1) $(-7.5,-6.5)$
(2) $(-7.5,-6.5]$
(3) $[-7.5,-6.5]$
(4) $[-7.5,-6.5)$
Show Answer
Answer: (1)
Solution:
Formula: Properties of greatest integer function, Standard result of limits
$\lim _{x \to a}([x-5]-[2 x+2])=0$
$\lim _{x \to a}([x]-5-[2 x]-2)=0$
$\lim _{x \to a}([x]-[2 x])=7$
$[a]-[2 a]=7$
$a \in I, a=-7$
$a \notin I, a=I+f$
Now, $[a]-[2 a]=7$
$ -I-[2 f]=7 $
Case-I: $f \in(0, \frac{1}{2})$
2f $\in(0,1)$
$-I=7$
$I=-7 \Rightarrow a \in(-7,-6.5)$
Case-II: $f \in(\frac{1}{2}, 1)$
2f $\in(1,2)$
$-I-1=7$
$I=-8 \Rightarrow a \in(-7.5,-7)$
Hence, $a \in(-7.5,-6.5)$
