Q99 / 160 · AP EAMCET 2022 (08 Jul Shift 2) · Mathematics

If [ $\cdot]$ denotes greatest integer function, then $\lim _{x \rightarrow \frac{\pi^{+}}{2}} \frac{[\sin x]-[\cos x]+1}{2}=$

  • A
    $0$
  • B
    $\frac{-1}{2}$
  • C
    $\frac{1}{2}$
  • Correct answer
    D
    $1$

Solution

$\begin{aligned} & \lim _{x \rightarrow \frac{\pi^{+}}{2}} \frac{[\sin x]-[\cos x]+1}{2} \\ & =\frac{0-(-1)+1}{2} \quad\left[\begin{array}{l}\text { For } x>\frac{\pi}{2}, 0 \leq \sin x < 1 \\ \text { and }-1 \leq \cos x < 0\end{array}\right] \\ & =\frac{2}{2}=1\end{aligned}$

Hence, the correct answer is option: 'D'.

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