Question #41717

For sufficiently small $x > 0$, let $A_x$ and $B_x$ be two events such that $P(A_x) = x$ and $P(B_x) = x^2$. The probability of their union satisfies the integral equation: $\int_0^x t P(A_t \cup B_t) dt = \frac{x^3}{3} + \frac{x^4}{4} - \frac{153}{5} x^5 - \int_0^x \frac{x^2 - t^2}{2} t^2 \sin\left(\frac{1}{t}\right) dt$ Let $L = \lim_{x \to 0^+} \frac{P(A_x \cap B_x)}{P(A_x) P(B_x)}$. Find the value of $L$.