Question

Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = ______.

Options

• `2/3`

• `3/2`

• 3

• 1

Answer 1

Let f : R `rightarrow` R be a positive increasing function with `lim_(x rightarrow ∞) (f(3x))/(f(x))` = 1 then `lim_(x rightarrow ∞) (f(2x))/(f(x))` = 1.

Explanation:

Given that f(x) is a positive increasing function.

∴ 0 < f(x) < f(2x) < f(3x)

Divided by f(x)

⇒ `0 < 1 < (f(2x))/(f(x)) < (f(3x))/(f(x))`

⇒ `lim_(x rightarrow ∞) 1 ≤ lim_(x rightarrow ∞) (f(2x))/(f(x)) ≤ lim_(x rightarrow ∞) (f(3x))/(f(x))`

By Sandwich Theorem,

⇒ `lim_(x rightarrow ∞) (f(2x))/(f(x))` = 1