Question

The value of $\underset{x\to \pi /2}{\lim }\left(\sec x-\tan x\right)$ is

(a) 2

(b) −1

(c) 1

(d) 0

Answer 1

(d) 0

$$\begin{gathered} \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left(\sec x-\tan x\right) \\ =\underset{h\to 0}{\lim }\left(\sec \left(\frac{\mathrm{\pi }}{2}-h\right)-\tan \left(\frac{\mathrm{\pi }}{2}-h\right)\right) \\ =\underset{h\to 0}{\lim }\left(\mathrm{cosec}h-\cot h\right) \\ =\underset{h\to 0}{\lim }\frac{1-\cos h}{\sin h} \\ =\underset{h\to 0}{\lim }\frac{2{\sin }^2{\displaystyle \frac{h}{2}}}{\sin h} \\ =\underset{h\to 0}{\lim }\frac{2{\sin }^2{\displaystyle \frac{h}{2}}}{2\sin {\displaystyle \frac{h}{2}}\cos {\displaystyle \frac{h}{2}}} \\ =\underset{h\to 0}{\lim }\tan \frac{h}{2} \\ =0 \end{gathered}$$