The set of all values of a for which [ _x a ([x - 5] - [2x + 2]) = 0], where [[]] denotes …

Question

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

Accepted Answer

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

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

$$ \Rightarrow \mathop {\lim }\limits_{x \to a} [x] - [2x] - 7 = 0$$

Case 1 :

If $$a \in \left[ {n,n + {1 \over 2}} \right)$$

then $$\mathop {\lim }\limits_{x \to a} \left( {[x] - [2x] - 7} \right) = 0$$

$$ \Rightarrow n - 2n - 7 = 0$$

$$ \Rightarrow n =  - 7$$

$$\therefore$$ $$a \in \left[ { - 7, - 6.5} \right)$$

Case 2 :

If $$a \in \left[ {n + {1 \over 2},n + 1} \right)$$

then $$\mathop {\lim }\limits_{x \to a} \left( {[x] - [2x] - 7} \right) = 0$$

$$ \Rightarrow n - (2n + 1) - 7 = 0$$

$$ \Rightarrow  - n - 8 = 0$$

$$ \Rightarrow n =  - 8$$

$$\therefore$$ $$a \in \left[ { - 7.5, - 7} \right)$$

$$R.H.L = \mathop {\lim }\limits_{x \to  - {{7.5}^ + }} \left( {\left[ x \right] - \left[ {2x} \right] - 7} \right)$$

$$ =  - 8 - \left( { - 15} \right) - 7$$

$=  - 8 + 15 - 7 = 0$

$$\eqalign{
  \& L.H.L = \mathop {\lim }\limits_{x \to  - {{7.5}^ - }} \left( {\left[ x \right] - \left[ {2x} \right] - 7} \right)  \cr \
  \&  =  - 8 - \left( { - 16} \right) - 7  \cr \
  \&  =  - 8 + 16 - 7 = 1 \cr} $$

$$ \therefore $$ R.H.L $$ \ne $$ L.H.L

$$ \therefore $$ At x = -7.5, limit does not exists.

$$\therefore$$ $$a \in \left( { - 7.5, - 6.5} \right)$$

Solutions · 1