Let $f_1 : \mathbb{R} \to \mathbb{R}$, $f_2 : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to \mathbb{R}$, $f_3 : \left(-1, e^{\frac{\pi}{2}} - 2\right) \to \mathbb{R}$ and $f_4 : \mathbb{R} \to \mathbb{R}$ be defined by \ (i) $f_1(x) = \sin\left(\sqrt{1 - e^{-x^2}}\right)$ \ (ii) $f_2(x) = \begin{cases} \dfrac{|\sin x|}{\tan^{-1} x} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$, where the inverse trigonometric function $\tan^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ \ (iii) $f_3(x) = \left[\sin\left(\log_e (x + 2)\right)\right]$, where, for $t \in \mathbb{R}$, $[t]$ denotes the greatest integer less than or equal to $t$ \ (iv) $f_4(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases}$ | List-I | List-II | | --- | --- | | (P) The function $f_1$ is | (1) NOT continuous at $x = 0$ | | (Q) The function $f_2$ is | (2) continuous at $x = 0$ and NOT differentiable at $x = 0$ | | (R) The function $f_3$ is | (3) differentiable at $x = 0$ and its derivative is NOT continuous at $x = 0$ | | (S) The function $f_4$ is | (4) differentiable at $x = 0$ and its derivative is continuous at $x = 0$ | The correct option is:

Let f₁ : R → R, f₂ : (-π/2, π/2) → R, f₃ : (-1, e^(π/2) - 2) → R and f₄ : R → R be defined by (i) f₁(x) = sin(√1 - e^(-x²)) (ii) f₂(x) = (|sin x|)/(tan⁻¹ x) if x ≠ 0 1 if x = 0 , where the inverse trigonometric function tan⁻¹ x assumes values in (-π/2, π/2) (iii) f₃(x) = [

$f_1(x) = \sin\!\left(\sqrt{1 - e^{-x^2}}\right)$

$f_2(x) = \dfrac{|\sin x|}{\tan^{-1}x}\ (x\neq 0),\quad f_2(0)=1$

$f_3(x) = \left[\sin(\ln(x+2))\right],\qquad f_4(x) = x^2\sin\tfrac1x\ (x\neq0),\ f_4(0)=0$

$\text{Match: } P(f_1),\ Q(f_2),\ R(f_3),\ S(f_4) \text{ with their behaviour at } x=0$

$1 - e^{-x^2} \sim x^2 \Rightarrow \sqrt{1-e^{-x^2}} \sim |x|$

$f_1(x) \sim \sin|x| \sim |x|:\ \text{continuous, slopes } \mp1 \Rightarrow \text{not differentiable}$

$\Rightarrow\ P \to 2$

$\dfrac{|\sin x|}{\tan^{-1}x} \sim \dfrac{|x|}{x} \to -1\ (\text{left}),\ +1\ (\text{right})$

$\text{limit does not exist} \Rightarrow \text{not continuous at } 0$

$\Rightarrow\ Q \to 1$

$\sin(\ln 2) \approx 0.639 \ \in (0,1)$

$\text{near } 0:\ [\sin(\ln(x+2))] = 0 \Rightarrow f_3 \text{ locally constant}$

$\Rightarrow\ R \to 4$

$|f_4(x)| \le x^2 \Rightarrow \text{continuous};\quad f_4'(0) = \lim_{x\to0} x\sin\tfrac1x = 0$

$f_4'(x) = 2x\sin\tfrac1x - \cos\tfrac1x\ (x\neq0):\ \text{oscillates} \Rightarrow f' \text{ not continuous}$

$\Rightarrow\ S \to 3$

$P \to 2,\quad Q \to 1,\quad R \to 4,\quad S \to 3$

$\boxed{\text{option } (4)}$