The set of all values of a for which $\underset{x\to a}{\mathrm{Lim}}\left(\right[x-5]-[2x+2\left]\right)=0$, where $\left[k\right]$  denotes the greater integer less than or equal to $k$  is equal to

$\underset{x\to a}{\lim }\left(\right[x-5]-[2x+2\left]\right)=0$

$\underset{x\to a}{\lim }\left(\right[x]-5-[2x]-2)=0$

$\underset{x\to a}{\lim }\left(\right[x]-[2x\left]\right)=7$

$\left[a\right]-\left[2a\right]=7$

$a\in I,a=-7$

$\text{a}\notin \text{I},\text{\hspace{1em}a}=\text{I}+\text{f}$

$\text{Now, }\left[a\right]-\left[2a\right]=7$

$-\text{I}-\left[2\text{f}\right]=7$

$Case-I:\text{f}\in (0,\frac{1}{2})$

$2\text{f}\in (0,1)$

$-\text{I}=7$

$\text{I}=-7\Rightarrow \text{a}\in (-7,-6.5)$

Case-II: $\text{f}\in (\frac{1}{2},1)$

$\begin{array}{l}\text{2f}\in (1,2)\\ -I-1=7\\ I=-8\Rightarrow a\in (-7.5,-7)\\ Hence,a\in (-7.5,-6.5)\end{array}$