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The function f x = l i m n → ∞ c o s 2 n π x + x is (where, . denotes the greatest integer function and n ∈ N )

Options

  1. Acontinuous at x = 1 but discontinuous at x = 3 2
  2. Bcontinuous at x = 1 and x = 3 2
  3. Cdiscontinuous at x = 1 and x = 3 2
  4. Ddiscontinuous at x = 1 but continuous at x = 3 2

Correct answer

D. discontinuous at x = 1 but continuous at x = 3 2

Step-by-step solution

f x = l i m n → ∞ c o s 2 π x n + x = x : cos 2 π x ∈ 0 ,1 1 + x : cos 2 π x = 1 = x : x ∉ Ι 1 + x : x ∈ Ι For x = 1 , f 1 - = 1 - = 0 ,   f 1 + = 1 + = 1 ,   f 1 = 1 + 1 = 2 ∵ f 1 + ≠ f 1 - ⇒ f x is discontinuous at x = 1 For x = 3 2 , f 3 2 + = 3 2 + = 1 ,   f 3 2 - = 3 2 - = 1 ,   f 3 2 = 3 2 = 1 ∵ f 3 2 - = f 3 2 + = f 3 2 ⇒ f x is continuous at x = 3 2

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