Given-f-R-R-is-increasing-positive-function-with-lim-x-f-3x-f-x-1-What-the-value-of-lim-x-f-2x-f-x-A-3-B-3-2-C-1-D-2-3-E-

Question Number 153765 by liberty last updated on 10/Sep/21

G i v e n f : R \to R i s i n c r e a s i n g p o s i t i v e

f u n c t i o n w i t h \lim_{x \to \infty} \frac{f (3 x)}{f (x)} = 1 .

W h a t t h e v a l u e o f \lim_{x \to \infty} \frac{f (2 x)}{f (x)} .

(A) 3 (B) \frac{3}{2} (C) 1 (D) \frac{2}{3} (E) \infty

Answered by gsk2684 last updated on 10/Sep/21

f (x) ⩽ f (2 x) ⩽ f (3 x)

1 ⩽ \frac{f (2 x)}{f (x)} ⩽ \frac{f (3 x)}{f (x)}

a s x \Rightarrow \infty, 1 ⩽ \frac{f (2 x)}{f (x)} ⩽ 1

a s x \Rightarrow \infty, \frac{f (2 x)}{f (x)} \Rightarrow 1

Commented by liberty last updated on 10/Sep/21

t h a n k y o u

Answered by puissant last updated on 10/Sep/21

\forall x \in R,

x ⩽ 2 x ⩽ 3 x \Rightarrow f (x) ⩽ f (2 x) ⩽ f (3 x)

(b e c a u s e f) .

\Rightarrow 1 ⩽ \frac{f (2 x)}{f (x)} ⩽ \frac{f (3 x)}{f (x)} (f (x) \neq 0) . .

\Rightarrow 1 ⩽ \lim_{x \to \infty} \frac{f (2 x)}{f (x)} ⩽ \lim_{x \to \infty} \frac{f (3 x)}{f (x)} = 1

(G e n d a r m t h e o r e m) . .

\Rightarrow \lim_{x \to \infty} \frac{f (2 x)}{f (x)} = 1

A n s w e r : C)

Commented by liberty last updated on 10/Sep/21

G e n d a r m t h e o r e m = S e q u e e z e t h e o r e m ?

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