Question

$\underset{x\to \pi /4}{\lim }\frac{\tan x-1}{\cos 2x}$ is equal to

• A. $1$
• B. $0$
• C. $-2$
• D. $-1$

Answer 1

Answer: (D)

$\underset{x\to \pi /4}{\lim }\frac{\tan x-1}{\cos 2x}$
$=\underset{h\to 0}{\lim }\frac{\tan \left(\frac{\pi }{4}+h\right)-1}{\cos 2\left(\frac{\pi }{4}+h\right)}$
$\left[\because x=\frac{\pi }{4}+h\right]$
$=\underset{h\to 0}{\lim }\frac{\left(\frac{1+\tan h}{1-\tan h}\right)-1}{\cos \left(\frac{\pi }{2}+2h\right)}$
$=\underset{h\to 0}{\lim }\frac{1+\tan h-1+\tan h}{-\sin 2h\left(1-\tan h\right)}$
$=\underset{h\to 0}{\lim }\frac{-2\tan h}{2\sin h\cos h\left(1-\tan h\right)}$
$=\underset{h\to 0}{\lim }\frac{-1}{{\cos }^{2}h\left(1-\tan h\right)}=-1$