Question

$\underset{\theta \to \pi /2}{\lim }\frac{1-\sin \theta }{\left(\pi /2-\theta \right)\cos \theta }$ is equal to

(a) 1

(b) −1

(c) $\frac{1}{2}$

(d) $-\frac{1}{2}$

Answer 1

(c) $\frac{1}{2}$

$$\begin{gathered} \underset{\mathrm{\theta }\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{1-\sin \mathrm{\theta }}{\left({\displaystyle \frac{\mathrm{\pi }}{2}}-\mathrm{\theta }\right)\cos \mathrm{\theta }} \\ =\underset{h\to 0}{\lim }\frac{1-\cos h}{\left({\displaystyle \frac{\mathrm{\pi }}{2}}-\left({\displaystyle \frac{\mathrm{\pi }}{2}}-h\right)\right)\sin h} \\ =\underset{h\to 0}{\lim }\frac{2{\sin }^2{\displaystyle \frac{h}{2}}}{h\sin h} \\ =\underset{h\to 0}{\lim }\frac{2{\sin }^2{\displaystyle \frac{h}{2}}}{{\displaystyle \frac{{\displaystyle \frac{4h^2}{4}}}{{\displaystyle \frac{\sin h}{h}}}}} \\ =\frac{2}{4} \\ =\frac{1}{2} \end{gathered}$$