Let f x = l i m n → ∞ x 2 + 2 x + 4 + sin ⁡ π x n - 1 x 2 + 2 x + 4 + sin ⁡ π x n + 1 , then

Question

Let f x = l i m n → ∞ x 2 + 2 x + 4 + sin ⁡ π x n - 1 x 2 + 2 x + 4 + sin ⁡ π x n + 1 , then, f x is continuous and differentiable for all x ∈ R ., f x is continuous but not differentiable for all x ∈ R ., f x is discontinuous at infinite number of points., f x is discontinuous at two points.

MARKS App · Accepted Answer

∵ x 2 + 2 x + 4 + s i n π x = x + 1 2 + 3 + sin ⁡ π x ≥ 2 , ∀ x ∈ R . ∴ f x = l i m n → ∞ 1 - x + 1 2 + 3 + sin ⁡ π x - n 1 + x + 1 2 + 3 + sin ⁡ π x - n = 1 - 0 1 + 0 = 1 Clearly, f x = 1 , ∀ x ∈ R Hence, f ( x ) is continuous and differentiable ∀ x ∈ R

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