Prove the following

Question:

$\lim _{x \rightarrow \pi / 4} \frac{\cot ^{3} x-\tan x}{\cos (x+\pi / 4)}$ is :

  1. 4

  2. $8 \sqrt{2}$

  3. 8

  4. $4 \sqrt{2}$


Correct Option: , 3

Solution:

$\lim _{x \rightarrow \pi / 4} \frac{\cot ^{3} x-\tan x}{\cos \left(x+\frac{\pi}{4}\right)}$

$\lim _{x \rightarrow \pi / 4} \frac{\left(1-\tan ^{4} x\right)}{\cos (x+\pi / 4)}$

$2 \lim _{x \rightarrow \pi / 4} \frac{\left(1-\tan ^{2} x\right)}{\cos (x+\pi / 4)}$

$R \lim _{x \rightarrow \pi / 4} \frac{\cos ^{2} x-\sin ^{2} x}{\frac{\cos x-\sin x}{\sqrt{2}}} \frac{1}{\cos ^{2} x}$

$4 \sqrt{2} \lim _{x \rightarrow \pi / 4}(\cos x+\sin x)=8$

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