Let f 1 : R → R, f 2 : , 
, f 3 : 
and f 4 : R → R be functions defined by
(i) 
(ii) 
, where the inverse trigonometric function tan –1 x assumes values in 
(iii) f 3 (x) = [sin(log e (x+2))], where for t ∈ R, [t] denotes the greatest integer less than or equal to t,
(iv) 
LIST-I LIST-II
(P) The function f 1 is
Text Solution
Verified by ExpertsD
Sol. (i) f ' 1 (0) = 
= 
= 1 × 1 × 
= 1 × 1 × 
= limit does not exist.
⇒ for option (P), is correct.
(ii) 
= 
= 
= limit does not exist ⇒ for option Q, is correct.
(iii) 
= 
now at x tends to zero (x + 2) tends to 2
⇒ log e (x + 2) tends to n2
⇒ log e (x + 2) → n2
which is less than 1
0 < 
sin(log e (x + 2)) < sin1 ⇒ 
[sin(log e (x + 2))] = 0
f 3 (x) = {0 x ∈ [–1, e π /2 – 2)
⇒ f ' 3 (x) = 0 ∀ x ∈ (–1, e π /2 – 2)
⇒ f " 3 (x) = 0 ∀ x ∈ (–1, e π /2 – 2)
Hence for (R), is correct.
(iv) 
f 4 (x) = 
= 
x 2 
= 0
f ' 4 (0) = 
= 
= 0
f ' 4 (x) = –cos 
+ xsin 
, x ≠ 0
f" 4 (0) = 
⇒ does not exist
hence for (S), is correct.
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