Let $x_1\geq ... \geq x_n = 0$ with $x = (x_1,...,x_n)$, $X =\left\{(x_1,...,x_n) \in [0,1]^n: x_1 \geq ... \geq x_n\right\},$ and $F$ be a twice continuously differentiable function with $F'(0)>0$. Evaluate the following limit: \begin{equation} \lim_{x \to 0, \ x\in X} \frac{F\left(\int_{0}^{1} \frac{(1-x_1)(1-2z)}{1-x_1 + \sum_{m=2}^n m (x_{m-1}-x_m)z^{m-1}}dz\right)}{F\left(1-\int_{0}^{1} \frac{(1-x_1)}{1-x_1 + \sum_{m=2}^n m (x_{m-1}-x_m)z^{m-1}}dz\right)}. \end{equation}
Attempt: both the numerator and the denominator converge to zero. Therefore, I apply l'Hopital's rule. However, I still get zero divided by zero. I guess that I should apply l'Hopital's rule again, but that does not seem like a good approach.