The value of lim x ? 0 2 ( 1 - cos x cos 2 x cos 3 x 3 ? cos 10 x 10 x 2 ) is _______ . [2024]

Question

The value of $\lim_{x \to 0} 2 (\frac{1 - \cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \dots \sqrt[10]{\cos 10 x}}{x^{2}})$ is _______ . [2024]

Smart Achievers · Accepted Answer

(55)

$\lim_{x \to 0} 2 (\frac{1 - \cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \dots \sqrt[10]{\cos 10 x}}{x^{2}})$

Let $f = \cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \dots \sqrt[10]{\cos 10 x}$

$f = \cos x {(\cos 2 x)}^{1 / 2} {(\cos 3 x)}^{1 / 3} \dots {(\cos 10 x)}^{1 / 10}$

Taking log on both sides, we get

$\log f = \log \cos x + \frac{1}{2} \cos 2 x + \frac{1}{3} \cos 3 x + \dots + \frac{1}{10} \cos 10 x$

Differentiating w.r.t. $x$

$\frac{1}{f} \frac{d f}{d x} = - \tan x - \tan 2 x \dots - \tan 10 x$

$\frac{d f}{d x} = - f (\tan x + \tan 2 x \dots + \tan 10 x)$

Using L'Hospital's Rule

$\lim_{x \to 0} 2 \frac{(f (\tan x + \tan 2 x \dots + \tan 10 x))}{2 x}$ $(∵ f = 1)$

$= 1 + 2 + \dots + 10 = 55$

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