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

We want to find $$\lim_{x\to0}2\left(\frac{1-\cos x\sqrt{\cos2x}\sqrt[3]{\cos3x}\cdots\sqrt[10]{\cos10x}}{x^2}\right).$$

Set $$P=\prod_{k=1}^{10}(\cos kx)^{1/k},$$ so that the expression inside the limit becomes $$2\frac{1-P}{x^2}.$$ Taking natural logarithms gives $$\ln P=\sum_{k=1}^{10}\frac{1}{k}\ln(\cos kx).$$

For small $$x$$ one has the approximation $$\ln(\cos kx)\approx-\frac{k^2x^2}{2},$$ which implies
$$\ln P\approx -\frac{x^2}{2}\sum_{k=1}^{10}\frac{k^2}{k}=-\frac{x^2}{2}\sum_{k=1}^{10}k=-\frac{x^2}{2}\cdot55=-\frac{55x^2}{2}.$$

Exponentiating this approximation yields $$P\approx e^{-55x^2/2}\approx1-\frac{55x^2}{2}$$ for small $$x$$, and therefore
$$1-P\approx\frac{55x^2}{2}.$$

Substituting into the limit gives
$$2\cdot\frac{1-P}{x^2}\approx2\cdot\frac{\frac{55x^2}{2}}{x^2}=55.$$

Hence, the value of the limit is 55.