Question

$\displaystyle\lim_{x\to \frac{\pi}{4}} \frac{\cot^{3} x - \tan x}{\cos\left(x+ \frac{\pi}{4}\right)} $ is :

• A. 4
• B. $ 8 \sqrt{2}$
• C. 8
• D. $4 \sqrt{2}$

Answer 1

Answer: (C)

$\lim_{x\to\pi/4} \frac{\cot^{3}x -\tan x}{\cos\left(x+ \frac{\pi}{4}\right)} $
$ \lim_{x\to\pi/4} \frac{\left(1-\tan^{4}x\right)}{\cos\left(x +\pi/4\right)} $
$ 2 \lim_{x\to\pi} \frac{\left(1-\tan^{4}x\right)}{\cos\left(x+\pi/4\right)} $
$R \lim_{x\to\pi/4} \frac{\cos^{2}x -\sin^{2}x}{\frac{\cos x -\sin x}{\sqrt{2}} } \frac{1}{\cos^{2}x} $
$4 \sqrt{2} \lim_{x\to\pi/4} \left(\cos x+\sin x\right) = 8 $