Question

The $\underset{x\to \frac{\pi }{2}}{\lim }\left\{2x\tan x-\frac{\pi }{\cos x}\right\}$ is

• A. -1
• B. -3
• C. -2
• D. 0

Answer 1

Answer: (C)

Let $L=\underset{x\to \frac{\pi }{2}}{\lim }\left\{2x\tan x-\frac{\pi }{\cos X}\right\}$
$=\underset{x\to \frac{\pi }{2}}{\lim }\left\{\frac{2x\sin x-\pi }{\cos x}\right\}$
$=\underset{x\to \frac{\pi }{2}}{\lim }\left\{2x\frac{\sin x}{\cos X}-\frac{\pi }{\cos x}\right\}$
$[\frac{0}{0}$ form ]
Now, by using L'Hospital Rule
$L=\underset{x\to \frac{\pi }{2}}{\lim }\left\{\frac{2\sin x+2x\cos x}{(-\sin x)}\right\}$
$=\frac{2\times 1+2\times \frac{\pi }{2}\times \cos \frac{\pi }{2}}{(-1)}=-2$