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

A

$$16$$

B

$$2$$

C

$$1$$

D

$$4$$