Given $z\geq 0$, denote $$A_m(z) = \{x\in \mathbf R^{m-1}\, :\, \min_{1\leq i\leq m-1} x_i > z\},$$ and $$F_m(z) = \int_{A_m(z)} (1+|x|^2)^{1-m} dx.$$ Does the following limit $$\lim\limits_{m\to\infty}\sqrt{m}(m-2)\int_0^1 (1-z)^{n-3}\frac{F_m(z)}{F_m(0)} dz$$ exist?

It is interesting if we know the value of this limit. If anyone know the references for this problem, please let me know.

Thanks in advances

Given $z\geq 0$, denote $$A_m(z) = \{x\in \mathbf R^{m-1}\, :\, \min_{1\leq i\leq m-1} x_i > z\},$$ and $$F_m(z) = \int_{A_m(z)} (1+|x|^2)^{1-m} dx.$$ Does the following limit $$\lim\limits_{m\to\infty}\sqrt{m}(m-2)\int_0^1 (1-z)^{m-3}\frac{F_m(z)}{F_m(0)} dz$$ exist?

It is interesting if we know the value of this limit. If anyone know the references for this problem, please let me know.

Thanks in advances

Given $z\leq 0$, denote $$A_m(z) = \{x\in \mathbf R^{m-1}\, :\, \min_{1\leq i\leq m-1} x_i > z\},$$ and $$F_m(z) = \int_{A_m(z)} (1+|x|^2)^{1-m} dx.$$ Does the following limit $$\lim\limits_{m\to\infty}\sqrt{m}(m-2)\int_0^1 (1-z)^{n-3}\frac{F_m(z)}{F_m(0)} dz$$ exist?

It is interesting if we know the value of this limit. If anyone know the references for this problem, please let me know.

Thanks in advances

Given $z\geq 0$, denote $$A_m(z) = \{x\in \mathbf R^{m-1}\, :\, \min_{1\leq i\leq m-1} x_i > z\},$$ and $$F_m(z) = \int_{A_m(z)} (1+|x|^2)^{1-m} dx.$$ Does the following limit $$\lim\limits_{m\to\infty}\sqrt{m}(m-2)\int_0^1 (1-z)^{n-3}\frac{F_m(z)}{F_m(0)} dz$$ exist?

It is interesting if we know the value of this limit. If anyone know the references for this problem, please let me know.

Thanks in advances