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

To solve the problem, we need to find the value of:
\[ \lim_{x \to 0} 2 \left( \frac{1 - \cos x \sqrt{\cos 2x} \, \sqrt[3]{\cos 3x} \ldots \sqrt[10]{\cos 10x}}{x^2} \right) \]
We start by analyzing the expression inside the cosine product:
Expression: \(\cos x \sqrt{\cos 2x} \sqrt[3]{\cos 3x} \ldots \sqrt[10]{\cos 10x}\)
Let's denote this product as \(P(x)\). Using the exponential form of cosine, this becomes:
\(P(x) = \cos x \cdot (\cos 2x)^{1/2} \cdot (\cos 3x)^{1/3} \cdot \ldots \cdot (\cos 10x)^{1/10}\)
The approximation for small angles \(x\) is \(\cos kx \approx 1 - \frac{(kx)^2}{2}\). Applying this:
\(P(x) \approx 1 - \frac{x^2}{2} - \frac{(2x)^2}{4} - \frac{(3x)^2}{6} - \ldots - \frac{(10x)^2}{20}\)
Summing these series terms gives:
\(P(x) \approx 1 - x^2 \left(\frac{1}{2} + \frac{4}{4} + \frac{9}{6} + \ldots + \frac{100}{20}\right)\)
This transforms into a sum of \(P_k = \frac{k^2}{2k}\), simplifying to:
\(P(x) \approx 1 - x^2 \left( \frac{1}{2} + 1 + 1.5 + \ldots + 5 \right)\)
Calculating the arithmetic series, we have:
\(S = \frac{1}{2} + 1 + 1.5 + \ldots + 5 = 27.5\)
Thus, \(P(x) \approx 1 - 27.5x^2\). Substituting back into our limit expression:
\[ \lim_{x \to 0} 2 \left(\frac{1 - (1 - 27.5x^2)}{x^2}\right) \]
This simplifies to:
\[ \lim_{x \to 0} 2 \times \frac{27.5x^2}{x^2} \]
Simplifying gives:
\[ \lim_{x \to 0} 2 \times 27.5 = 55 \]
The calculated value, 55, falls within the expected range of 55 to 55.

Answer 2

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