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

<p>$\mathop {\lim }\limits_{x \to a} \left( {[x - 5] - [2x + 2]} \right) = 0$</p> <p>$$ \Rightarrow \mathop {\lim }\limits_{x \to a} \left( {[x] - 5 - [2x] - 2} \right) = 0$$</p> <p>$\Rightarrow \mathop {\lim }\limits_{x \to a} [x] - [2x] - 7 = 0$</p> <p>Case 1 :</p> <p>If $a \in \left[ {n,n + {1 \over 2}} \right)$</p> <p>then $\mathop {\lim }\limits_{x \to a} \left( {[x] - [2x] - 7} \right) = 0$</p> <p>$\Rightarrow n - 2n - 7 = 0$</p> <p>$\Rightarrow n = - 7$</p> <p>$\therefore$ $a \in \left[ { - 7, - 6.5} \right)$</p> <p>Case 2 :</p> <p>If $a \in \left[ {n + {1 \over 2},n + 1} \right)$</p> <p>then $\mathop {\lim }\limits_{x \to a} \left( {[x] - [2x] - 7} \right) = 0$</p> <p>$\Rightarrow n - (2n + 1) - 7 = 0$</p> <p>$\Rightarrow - n - 8 = 0$</p> <p>$\Rightarrow n = - 8$</p> <p>$\therefore$ $a \in \left[ { - 7.5, - 7} \right)$</p> <p>$$R.H.L = \mathop {\lim }\limits_{x \to - {{7.5}^ + }} \left( {\left[ x \right] - \left[ {2x} \right] - 7} \right)$$</p> <p>$= - 8 - \left( { - 15} \right) - 7$</p> <p>$= - 8 + 15 - 7 = 0$</p> <p>$$\eqalign{ & L.H.L = \mathop {\lim }\limits_{x \to - {{7.5}^ - }} \left( {\left[ x \right] - \left[ {2x} \right] - 7} \right) \cr \\ & = - 8 - \left( { - 16} \right) - 7 \cr \\ & = - 8 + 16 - 7 = 1 \cr} $$</p> <p>$\therefore$ R.H.L $\ne$ L.H.L</p> <p>$\therefore$ At x = -7.5, limit does not exists.</p> <p>$\therefore$ $a \in \left( { - 7.5, - 6.5} \right)$</p>