Solve this following

Question:

For each $t \in R$, let [t] be the greatest integer less than or equal to $\mathrm{t}$. Then,

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

  1. equals $-1$

  2. equals 1

  3. does not exist

  4. equals 0

Correct Option: , 4

Solution:

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

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

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

 

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