QJEEM24Ma8d63: JEE Main 2024 Mathematics - Class 12 JEEM24Ma8d63 Solution: The value of lim x → 0 2 ( 1 − cos ⁡ x cos ⁡ 2 x cos ⁡ 3 x 3 ……

$\mathop {\lim }\limits_{x \to 0} 2\left( {{{1 - \cos x{{(\cos 2x)}^{{1 \over 2}}}{{(\cos 3x)}^{{1 \over 3}}}\,...\,{{(\cos 10x)}^{{1 \over {10}}}}} \over {{x^2}}}} \right)$ $\left(\frac{0}{0} \text { form }\right)$

Using L' hospital

$$2\underset{x\to 0}{lim}\frac{\sin x(\cos 2x{)}^{\frac{1}{2}}\dots (\cos 10x{)}^{\frac{1}{10}}\dots (\sin 2x)(\cos x)(\cos 3x{)}^{\frac{1}{3}}+\dots }{2x}$$

$\begin{aligned}\Rightarrow & \lim _{x \rightarrow 0}\left(\frac{\sin x}{x}+\frac{\sin 2 x}{x}+\ldots+\frac{\sin 10 x}{x}\right) \\\quad & =1+2+\ldots+10=55\end{aligned}$