For any positive integer n, define f n : (0, ∞ ) → R as
f n (x) = 
for all x ∈ (0, ∞ ). (Here, the inverse trigonometric function tan –1 x assumes values in 
)
Then, which of the following statement(s) is (are) TRUE ?
Text Solution
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Sol. f n (x) = 
f n (x) = tan –1 (x + n) – tan –1 (x) ⇒ f ' n (x) = 
f n (0) = tan –1 (n) ⇒ tan 2 (tan –1 n) = n 2
= 
f ' n (0) = 
⇒ 1 + f n (0) = 
sec 2 (f n (0)) = sec 2 (tan –1 (n)) = 1+ n 2 .
Hence vr% (1 + f ' n (0)).sec 2 (f n (0)) = 
(1 + n 2 ) = 1
so , 
sec 2 (f i (0)) = 
= 
(f n (x)) = 0 & 
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