Q85 / 180 · COMEDK 2013 · Mathematics

$\lim _{a \rightarrow \pi / 2} \frac{\frac{\pi}{2}-\theta}{\cot \theta}=$

  • A
    $0$
  • B
    $-1$
  • Correct answer
    C
    $1$
  • D
    $\infty$

Solution

We have, $$ \begin{aligned} &\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{\frac{\pi}{2}-\theta}{\cot \theta}=\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{-1}{-\operatorname{cosec}^{2} \theta} \text { [usingL'Hospital Rule] } \\ &\Rightarrow \lim _{\theta \rightarrow \frac{\pi}{2}} \sin ^{2} \theta=\sin ^{2} \frac{\pi}{2}=1 \end{aligned} $$

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

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