Q69 / 120 · VITEEE 2015 · Mathematics
$\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{\cos 2 x}$ is equal to
- A1
- B0
- C$-2$
- Correct answerD$-1$
Solution
$\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{\cos 2 x}=\lim _{h \rightarrow 0} \frac{\tan \left(\frac{\pi}{4}+h\right)-1}{\cos 2\left(\frac{\pi}{4}+h\right)}$
$=\left[\because x=\frac{\pi}{4}+h\right]$
$=\lim _{h \rightarrow 0} \frac{\left(\frac{1+\tan h}{1-\tan h}\right)-1}{\cos \left(\frac{\pi}{2}+2 h\right)}$
$=\lim _{h \rightarrow 0} \frac{1+\tan h-1+\tan h}{-\sin 2 h(1-\tanh )}$
$=\lim _{h \rightarrow 0} \frac{-2 \tan h}{2 \sin h \cos h(1-\tan h)}$
$=\lim _{h \rightarrow 0} \frac{-1}{\cos ^{2} h(1-\tan h)}=-1$
Hence, the correct answer is option: 'D'.