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For any positive integer n, define ƒ n : (0, ∞ ) → R as ƒ n (x) = n 1 j = Σ tan –1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + + + ) 1 j x )( j x ( 1 1 for all x ∈ (0, ∞ ). ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π π − . 2 , 2 in values assumes x tan function ric trigonomet inverse the Here, 1 – Then, which of the following statement(s) is (are) TRUE ?

1. 5 1 j = Σ tan 2 (ƒ j (0)) = 55
2. 10 1 j = Σ (1 + j ƒ ′ (0)) sec 2 (ƒ j (0)) = 10
3. For any fixed positive integer n, ∞ → x lim tan (ƒ n (x)) = n 1
4. For any fixed positive integer n, ∞ → x lim sec 2 (ƒ n (x)) = 1

Correct answer: For any fixed positive integer n, ∞ → x lim sec 2 (ƒ n (x)) = 1

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