### Limits Question 2

#### Question 2 - 24 January - Shift 2

The set of all values of a for which

$lim _{x \to a}([x-5]-[2 x+2])=0$, where $[a]$ denotes the greater integer less than or equal to $a$ is equal to

(1) $(-7.5,-6.5)$

(2) $(-7.5,-6.5]$

(3) $[-7.5,-6.5]$

(4) $[-7.5,-6.5)$

## Show Answer

#### Answer: (1)

#### Solution:

#### Formula: Properties of greatest integer function, Standard result of limits

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

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

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

$[a]-[2 a]=7$

$a \in I, a=-7$

$a \notin I, a=I+f$

Now, $[a]-[2 a]=7$

$ -I-[2 f]=7 $

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

2f $\in(0,1)$

$-I=7$

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

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

2f $\in(1,2)$

$-I-1=7$

$I=-8 \Rightarrow a \in(-7.5,-7)$

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