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

Correct Answer: 55

\[ \lim_{x \to 0} 2 \cdot \frac{\left( 1 - \frac{x^2}{2!} \right) \left( 1 - \frac{4x^2}{2!} \right) \left( 1 - \frac{9x^2}{2!} \right) \cdots \left( 1 - \frac{100x^2}{2!} \right)}{x^2} \]
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.