Let $S$ be the set of all $(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}$ such that

$$ \lim\limits_{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\log _e(1+x)\right)^\beta}=0 . $$

Then which of the following is (are) correct?

Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^2\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?

Let f : R $$\to$$ R be defined by $$f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}$$

Then which of the following statements is (are) TRUE?

Let f : R $$ \to $$ R and g : R $$ \to $$ R be functions
satisfying f(x + y) = f(x) + f(y) + f(x)f(y)
and f(x) = xg(x) for all x, y$$ \in $$R.
If $$\mathop {\lim }\limits_{x \to 0} g(x) = 1$$, then which of the following statements is/are TRUE?