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Step-by-Step Solution
Step 1: Express the given limit in a simpler form
We want to evaluate the expression
$ \lim_{x \to a} \bigl([x - 5] - [2x + 2]\bigr) $
and to find those values of $a$ for which this limit equals 0. Note that $[\alpha]$ is the greatest integer less than or equal to $\alpha$. We can rewrite
$[x - 5] - [2x + 2]$ as
$[x] - 5 - [2x] - 2$,
so our limit becomes
$ \lim_{x \to a} \bigl([x] - [2x] - 7\bigr) .$
Step 2: Understand the behavior of the greatest integer function
The greatest integer function $[x]$ “jumps” at integer points. Hence, to understand
$[x]$ and $[2x]$ near $x = a$, we consider intervals where the fractional parts of $x$ and $2x$ do not cross integer boundaries.
Step 3: Case 1 — When $a \in [n,\, n + 0.5)$ for some integer $n$
In this interval, $[x]$ remains $n$ as $x \to a$. Also, $2x$ remains in $[2n,\, 2n+1)$, so $[2x]$ will be $2n$. Thus, as $x \to a$ in $[n,\, n + 0.5)$,
$$
[x] = n, \quad [2x] = 2n.
$$
Therefore,
$$
[x] - [2x] - 7 = n - 2n - 7 = -n - 7.
$$
For the limit to equal 0, we need
$$
-n - 7 = 0 \quad \Longrightarrow \quad n = -7.
$$
Consequently, if $n = -7$, the interval for $a$ in this case is
$[-7,\, -6.5)$.
Step 4: Case 2 — When $a \in [n + 0.5,\, n + 1)$ for some integer $n$
Here, $[x]$ remains $n$, but $2x \in [2n+1,\, 2n+2)$, implying $[2x] = 2n + 1$. So,
$$
[x] - [2x] - 7 = n - (2n+1) - 7 = -n - 8.
$$
For the limit to be 0,
$$
-n - 8 = 0 \quad \Longrightarrow \quad n = -8.
$$
Therefore, in this situation, if $n = -8$, the interval for $a$ is
$[-7.5,\, -7)$.
Step 5: Check the continuity at critical points
The intervals we found are $[-7,\, -6.5)$ (from Case 1) and $[-7.5,\, -7)$ (from Case 2). We need to check what happens at the boundary points, especially at $-7.5$:
Right-hand limit as $x \to (-7.5)^+$:
In this small interval to the right of $-7.5$, $[x] = -8$ and $[2x] = -15$. Substituting:
$$
\lim_{x \to -7.5^+} ([x] - [2x] - 7)
= (-8) - (-15) - 7 = -8 + 15 - 7 = 0.
$$
Left-hand limit as $x \to (-7.5)^-$:
Just to the left of $-7.5$, $[x] = -8$ but $[2x] = -16$. Thus:
$$
\lim_{x \to -7.5^-} ([x] - [2x] - 7)
= (-8) - (-16) - 7 = -8 + 16 - 7 = 1.
$$
Because the left-hand limit ($1$) is not equal to the right-hand limit ($0$) at $x = -7.5$, the limit does not exist there. Thus we exclude $x = -7.5$.
Step 6: Combine the valid intervals
From Case 1 and Case 2, we have the points from $[-7.5, -7)$ and $[-7, -6.5)$. Combining and removing the point $-7.5$ where the limit does not exist (and also checking similarly for $-6.5$ if needed), the final interval on which the limit equals 0 is:
$$
(-7.5,\, -6.5).
$$
Final Answer
The set of all values of $a$ is
$(-7.5, -6.5)\,.$
