Let f_1:\mathbb{R}\to\mathbb{R},f_2:(-\pi/2,\pi/2)\to\mathbb{R},f_3(-1,e^{\pi/2}-2)\to\mathbb{R} and f_4:\mathbb{R}\to\mathbb{R} be functions defied by (i) f_1(x)=\sin(\sqrt{1-e^{-x^2}) (ii) f_2(x)=\{\table{|sinx|}/{\tan^{-1}x},if,x\neq0;1,if,x=0, where the inverse trigonometric function \tan^{-1}x assumes values in (-\pi/2,\pi/2) (iii) f_3(x)=[\sin(\text{\log}_e(x+2))], where for t∊\mathbb{R},[t] denotes the greatest integer less than or equal to t, (iv) f_4(x)=\{\table x^2sin(1/x),if,x\neq0;0,if,x=0 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 is 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 options is: P Q R S A) 2 3 1 4 B) 4 1 2 3 C) 4 2 1 3 D) 2 1 4 3 [JEE Adv 2018 P2]

Limits, Continuity and Differentiability

Section: Mathematics

Question : 22 of 51

Marks: +1, -0

Let

f1:ℝ→ℝ,f2:(−

)→ℝ,f3(−1,e

−2)→ℝ

and f4:ℝ→ℝ be functions defied by
(i) f1(x)=sin(√1−e−x2)
(ii) f2(x)={

|sinx|

tan−1x

x≠0

x=0

,
where the inverse trigonometric function tan−1x assumes values in (−

)
(iii) f3(x)=[sin(loge(x+2))], where for t∊ℝ,[t] denotes the greatest integer less than or equal to t,
(iv) f4(x)={

x2sin(

)

x≠0

x=0

List - I	List - II
P. the function f1 is	1. NOT continuous at x=0
Q. The function f2 is	2. Continuous at x=0 and NOT differentiable at x=0
R. The function f3 is	3. Differentiable at x=0 and is derivative is NOT continuous at x=0
S. The function f4 is	4. Differentiable at x=0 and its derivative is continuous at x=0

The correct options is:

[JEE Adv 2018 P2]

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