Question

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

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

Answer 1

Answer: (C)

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