Let \( S \) be the set of all \( (\alpha,\beta) \in \mathbb{R} \times \mathbb{R} \) such that
\( \displaystyle \lim_{x \to \infty} \frac{\sin(x^2)(\log_e x)^{\alpha} \sin\!\left(\frac{1}{x^2}\right)}{x^{\alpha\beta}\,(\log_e(1+x))^{\beta}} = 0 \)
The which of the following is (are) correct?
