Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by

$$ f(x)=\left\{\begin{array}{cc} x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0, \\ 0, & \text { if } x=0 . \end{array}\right. $$

Then which of the following statements is TRUE?

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?

A straight line drawn from the point $P(1,3,2)$, parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$, intersects the plane $L_1: x-y+3 z=6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L_1$ intersects the plane $L_2: 2 x-y+z=-4$ at the point $R$. Then which of the following statements is (are) TRUE?

Let $A_1, B_1, C_1$ be three points in the $x y$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8 x$ at $A_1$ and $B_1$, respectively. If $O=(0,0)$ and $C_1=(-4,0)$, then which of the following statements is (are) TRUE?