Step-by-step Solution:

We are given the limit:\n\[ L = \lim_{x \to 0} \frac{|x|\log_e(1+|\sin 2x|)}{x^2(|x|+3)} \]\n\nRecall the standard limit for logarithms: \( \lim_{u \to 0} \frac{\log_e(1+u)}{u} = 1 \).\nLet \( u = |\sin 2x| \). As \( x \to 0 \), \( u \to 0 \).\nTherefore, \( \log_e(1+|\sin 2x|) \approx |\sin 2x| \).\nAlso, for small \( x \), \( \sin 2x \approx 2x \), which means \( |\sin 2x| \approx |2x| = 2|x| \).\n\nSubstitute this approximation into the expression:\n\[ L = \lim_{x \to 0} \frac{|x| \cdot 2|x|}{x^2(|x|+3)} \]\n\[ = \lim_{x \to 0} \frac{2|x|^2}{x^2(|x|+3)} \]\n\nSince \( |x|^2 = x^2 \) for all real numbers:\n\[ = \lim_{x \to 0} \frac{2x^2}{x^2(|x|+3)} \]\n\[ = \lim_{x \to 0} \frac{2}{|x|+3} \]\n\nNow substitute \( x = 0 \) into the simplified limit:\n\[ L = \frac{2}{0 + 3} = \frac{2}{3} \]\n\nThe limit exists and is equal to \( \frac{2}{3} \).