The set of all values of a for which lim x a [ x - 5 ] - [ 2 x + 2 ] = 0 , where denotes the greatest integer less than or equal to is equal to [2023]

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Solution

Q.
$The set of all values of a for which \lim_{x \to a} ([x - 5] - [2 x + 2]) = 0,$ where $[α]$ denotes the greatest integer less than or equal to $α$ is equal to [2023]

1 $[- 7.5, - 6.5)$

2 $(- 7.5, - 6.5)$

3 $(- 7.5, - 6.5]$

4 $[- 7.5, - 6.5]$

Ans.
(2)

$\lim_{x \to a} ([x - 5] - [2 x + 2]) = 0$

$\Rightarrow \lim_{x \to a} ([x] - 5 - [2 x] - 2) = 0 \Rightarrow \lim_{x \to a} ([x] - [2 x]) = 7$

$Let a \in [x_{1}, x + \frac{1}{2}), then x - 2 x = 7 \Rightarrow x = - 7$

$∴$ $a \in [- 7, - 6.5)$

$Let a \in (x + \frac{1}{2}, x + 1), then x - (2 x + 1) = 7$

$\Rightarrow - x = 8 \Rightarrow x = - 8;$ $a \in [- 7.5, - 7)$

But limit does not exist at $a = - 7.5$

$Hence, a \in (- 7.5, - 6.5)$

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