Let $f:[-\tfrac{\pi}{2}, \tfrac{\pi}{2}] \to \mathbb{R}$ be a differentiable function such that $f(0)=\tfrac{1}{2}$. If $\displaystyle \lim_{x \to 0} \frac{x \int_0^x f(t),dt}{e^{x^2} - 1} = \alpha$, then $8\alpha^2$ is equal to: