Question

The value of $\lim_{x \to 0} 2 \left( \frac{1 - \cos x \sqrt{\cos 2x} \, \sqrt[3]{\cos 3x} \ldots \sqrt[10]{\cos 10x}}{x^2} \right)$ is _____.

Answer 1

$\lim_{x \to 0} 2 \cdot \frac{\left( 1 - \frac{x^2}{2!} \right) \left( 1 - \frac{4x^2}{2!} \right) \left( 1 - \frac{9x^2}{2!} \right) \cdots \left( 1 - \frac{100x^2}{2!} \right)}{x^2}$
By expansion:
$\lim_{x \to 0} 2 \cdot \frac{\left[ 1 - \frac{x^2}{2} \right] \left[ 1 - \frac{2x^2}{2} \right] \left[ 1 - \frac{3x^2}{2} \right] \cdots \left[ 1 - \frac{10x^2}{2} \right]}{x^2}.$
Simplify the product:
$\lim_{x \to 0} 2 \cdot \frac{1 - \left[ \frac{x^2}{2} + \frac{2x^2}{2} + \frac{3x^2}{2} + \cdots + \frac{10x^2}{2} \right]}{x^2}.$
The $x^2$ terms cancel:
$2 \cdot \left( \frac{1}{2} + \frac{2}{2} + \frac{3}{2} + \cdots + \frac{10}{2} \right).$
Simplify the summation:
$2 \cdot \frac{1 + 2 + 3 + \cdots + 10}{2}.$
The sum of the first 10 natural numbers is:
$\frac{10 \cdot 11}{2} = 55.$
Final Answer: 55.