function limit, Fourier transform, characteristic function

Question

I want to show the following function limit satisfies \begin{align} \mathop {\lim }\limits_{{x_m} \to 1} f({\bf x}) = 0\quad and \mathop {\lim }\limits_{{x_m} \to \infty} f({\bf x}) = 1 \end{align} where \begin{align} f({\bf x})=\int\limits_{0}^\infty {\int\limits_{ - \infty }^\infty {\prod\limits_{m = 0}^{M - 1} {\frac{1}{{\sqrt {1 - 2jx_m w} }}\exp ( - jwt)} \frac{{dw}}{{2\pi }}dt} } \end{align} Maybe we can use some tricky ways, maybe some bounds, do you have some opinions, thank you so much.