The set of all values of a for which lim_{x r | vedclass

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(D) We are given $\lim_{x \rightarrow a}(\lfloor x-5 \rfloor - \lfloor 2x+2 \rfloor) = 0$.This simplifies to $\lim_{x \rightarrow a}(\lfloor x \rfloor

Vedclass · Accepted Answer

$[-7.5, -6.5)$

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(D) We are given $\lim_{x \rightarrow a}(\lfloor x-5 \rfloor - \lfloor 2x+2 \rfloor) = 0$.This simplifies to $\lim_{x \rightarrow a}(\lfloor x \rfloor - 5 - \lfloor 2x \rfloor - 2) = 0$,which implies $\lim_{x \rightarrow a}(\lfloor x \rfloor - \lfloor 2x \rfloor) = 7$.For the limit to exist and equal $7$,the left-hand limit and right-hand limit must be equal.Let $a = I + f$,where $I$ is an integer and $f \in [0, 1)$.If $f \in [0, 0.5)$,then $\lfloor a \rfloor = I$ and $\lfloor 2a \rfloor = 2I$. The condition becomes $I - 2I = 7 \Rightarrow I = -7$. Thus $a \in [-7, -6.5)$.If $f \in [0.5, 1)$,then $\lfloor a \rfloor = I$ and $\lfloor 2a \rfloor = 2I + 1$. The condition becomes $I - (2I + 1) = 7$ $\Rightarrow -I = 8$ $\Rightarrow I = -8$. Thus $a \in [-7.5, -7)$.Combining these,$a \in [-7.5, -6.5)$.

The set of all values of $a$ for which $\lim_{x \rightarrow a}(\lfloor x-5 \rfloor - \lfloor 2x+2 \rfloor) = 0$,where $\lfloor \alpha \rfloor$ denotes the greatest integer less than or equal to $\alpha$,is equal to

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