Question #41934

Let $L_1$ and $L_2$ be two lines in 3D space with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ respectively, where: $a_1 = \lim_{x \to 0} \frac{2(\tan x - \sin x)}{x^3}, \quad b_1 = \lim_{x \to 0} \frac{6(\tan^{-1}x - x)}{x^3}, \quad c_1 = \lim_{x \to 0} \frac{6(x - \sin x)}{x^3}$ $a_2 = \lim_{x \to 0} \frac{24(\cos x - 1 + \frac{x^2}{2})}{x^4}, \quad b_2 = \lim_{x \to 0} \frac{3(\ln(1+x) - x + \frac{x^2}{2})}{x^3}, \quad c_2 = \lim_{x \to 0} \frac{6(e^x - 1 - x - \frac{x^2}{2})}{x^3}$ If $\theta$ is the acute angle between the lines $L_1$ and $L_2$, then the value of $\theta$ is: