Question

For each $t \in R$, let [$t$] be the greatest integer less than or equal to $t$. Then,
$\displaystyle\lim_{x\to1+} \frac{\left(1-\left|x\right| +\sin\left|1-x\right|\right) \sin \left(\frac{\pi}{2} \left[1-x\right]\right) }{\left|1-x\right|\left[1-x\right]} $

• A. equals -1
• B. equals 1
• C. does not exist
• D. equals 0

Answer 1

Answer: (D)

$\lim _{x \rightarrow 1^{+}} \frac{(1-|x|+\sin |1-x|) \sin \left(\frac{\pi}{2}[1-x]\right)}{|1-x|[1-x]}$
$=\lim _{x \rightarrow 1^{+}} \frac{(1-x)+\sin (x-1)}{(x-1)(-1)} \sin \left(\frac{\pi}{2}(-1)\right)$
$=\lim _{x \rightarrow 1^{+}}\left(1-\frac{\sin (x-1)}{(x-1)}\right)(-1)=(1-1)(-1)=0$