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  4. The set of all values of a for which displaystyle lim x arrow a([x-5]-[2 x+2])=0, where [ propto] denotes the greatest integer less than or equal to propto is equal to

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

JEE MainJEE Main 2023Limits and Derivatives

Solution:

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