Let f : 0 , 1 → ℝ be the function defined as f x = n if x ∈ [ 1 n + 1 , 1 n ) where n ∈ ℕ . Let g : 0 , 1 → ℝ be a function such that ∫ x 2 x 1 - t t d t < g x < 2 x for all x ∈ 0 , 1 . Then lim x → 0 f x g x

Step-by-step solution

Given, f : 0 ,   1 → ℝ be the function defined as f x = n if x ∈ [ 1 n + 1 ,   1 n ) where n ∈ ℕ And g : 0 ,   1 → ℝ be a function such that ∫ x 2 x 1 - t t d t < g x < 2 x for all x ∈ 0 ,   1 . Now we need to solve 1 sided limit here to get some answer, as lim x → 0 - doesn’t exist here (not in domain) As 1 n + 1 ≤ x <   1 n ⇒ n + 1 ≥ 1 x >   n ⇒ n ≥ 1 x - 1 ⇒ n &#8805