Question

$\underset{\theta \to \frac{\pi }{2}}{\lim }\left(\frac{{\displaystyle \frac{\pi }{2}}-\theta }{\cot \theta }\right)=$

• A. $0$
• B. $-1$
• C. $1$
• D. $\infty$

Answer 1

The correct option is C

$1$

Explanation for the correct answer:

Finding the value of the given limit:

$\underset{\theta \to \frac{\pi }{2}}{\lim }\left(\frac{{\displaystyle \frac{\pi }{2}}-\theta }{\cot \theta }\right)$

Applying L'Hospital Rule, i.e., $\underset{x\to c}{\lim }\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\underset{x\to c}{\lim }\left(\frac{f\text{'}\left(x\right)}{g\text{'}\left(x\right)}\right)$

$$\begin{gathered} \underset{\theta \to \frac{\pi }{2}}{\lim }\left(\frac{{\displaystyle \frac{\pi }{2}}-\theta }{\cot \theta }\right) \\ =\underset{\theta \to \frac{\pi }{2}}{\lim }\left(\frac{-1}{-{\csc }^2\theta }\right)\left[\frac{d}{\mathrm{d\theta }}\left(\frac{\pi }{2}-\theta \right)=-1;\frac{d}{\mathrm{d\theta }}\left(\cot \theta \right)=-{\csc }^2\theta \right] \\ =\underset{\theta \to \frac{\pi }{2}}{\lim }\left(\frac{1}{{\csc }^2\theta }\right) \end{gathered}$$

Applying the limits,

$$\begin{gathered} =\frac{1}{1}\left[\because \csc \left(\frac{\pi }{2}\right)=1\right] \\ =1 \end{gathered}$$

Therefore, the correct answer is option (C).