Question

The value of $\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\sin \left(\cos \left(x\right)\right)\cos \left(x\right)}{\sin \left(x\right)-\csc \left(x\right)}$ is

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

Answer 1

The correct option is D

$-1$

Explanation of the correct option.

Compute the required value.

Given :$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\sin \left(\cos \left(x\right)\right)\cos \left(x\right)}{\sin \left(x\right)-\csc \left(x\right)}$

$$\begin{gathered} \Rightarrow \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\sin \left(\cos \left(x\right)\right)}{\cos \left(x\right)}\times \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{{\cos }^2\left(x\right)}{\sin \left(x\right)-{\displaystyle \frac{1}{\sin \left(x\right)}}} \\ \Rightarrow 1\times \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{{\cos }^2\left(x\right)\sin \left(x\right)}{{\sin }^2\left(x\right)-1} \\ \Rightarrow \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{{\cos }^2\left(x\right)\sin \left(x\right)}{-{\cos }^2\left(x\right)} \\ \Rightarrow \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }-\sin \left(x\right) \\ \Rightarrow -1 \end{gathered}$$

Therefore, the value of $\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\sin \left(\cos \left(x\right)\right)\cos \left(x\right)}{\sin \left(x\right)-\csc \left(x\right)}$ is $-1$.

Hence,option $\left(D\right)$ is the correct option.