JEE PYQ: Limits Question 35
Question 35 - 2019 (12 Jan Shift 2)
$\lim_{x \to \pi/4} \frac{\cot^3 x - \tan x}{\cos\left(x + \frac{\pi}{4}\right)}$ is:
(1) $4$
(2) $4\sqrt{2}$
(3) $8\sqrt{2}$
(4) $8$
Show Answer
Answer: (4)
Solution
$= \lim_{x\to\pi/4}\frac{\cot^3 x\left(1 - \frac{\tan x}{\cot^3 x}\right)}{\cos(x+\pi/4)} = \lim_{x\to\pi/4}\frac{(1-\tan^4 x)}{\tan^3 x \cdot \cos(x+\pi/4)} = \lim_{x\to\pi/4}\frac{(1+\tan^2 x)(1+\tan x)(1-\tan x)}{\tan^3 x \cdot \frac{\cos x - \sin x}{\sqrt{2}}} = \frac{(2)(2)}{\sqrt{2} \cdot \frac{1}{\sqrt{2}}} = 8$.
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