Let f ( x ) = x sin x , x ∈ ( 0 , 1 ) 1 , x = 0 , Consider the integral I n = n ∫ 0 1 / n f ( x ) e - n x d x Then lim n → ∞ I n

Question

Let f ( x ) = x sin x , x ∈ ( 0 , 1 ) 1 , x = 0 , Consider the integral I n = n ∫ 0 1 / n f ( x ) e - n x d x Then lim n → ∞ I n, does not exist, exists and is 0, exists and is 1, exists and is 1 - e - 1

MARKS App · Accepted Answer

f ( x ) is an increasing function. so, f ( x ) ∈ 1 , 1 sin 1 ∀ x ∈ [ 0 , 1 ) Now, n ∫ 0 1 / n e - n x d x ≤ n ∫ 0 1 / n f ( x ) e - n x d x ≤ n sin 1 ∫ 0 1 / n e - n x d x ⇒ lim n → ∞ 1 - 1 e n ≤ lim n → ∞ I n ≤ 1 - 1 e ( sin 1 ) n ⇒ 0 ≤ lim n → ∞ I n ≤ 0 ⇒ lim n → ∞ I n = 0

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