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

 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) ∞

$$\:{Given}\:{f}:{R}\rightarrow{R}\:{is}\:{increasing}\:{positive} \\ $$$${function}\:{with}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{3}{x}\right)}{{f}\left({x}\right)}=\mathrm{1}\:.\: \\ $$$${What}\:{the}\:{value}\:{of}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}. \\ $$$$\left({A}\right)\:\mathrm{3}\:\:\:\:\:\left({B}\right)\:\frac{\mathrm{3}}{\mathrm{2}}\:\:\:\:\:\left({C}\right)\:\mathrm{1}\:\:\:\:\:\left({D}\right)\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\left({E}\right)\:\infty \\ $$

Answered by gsk2684 last updated on 10/Sep/21

f(x)≤f(2x)≤f(3x)  1≤((f(2x))/(f(x)))≤((f(3x))/(f(x)))  as x⇒∞ , 1≤((f(2x))/(f(x)))≤1  as x⇒∞, ((f(2x))/(f(x)))⇒1

$${f}\left({x}\right)\leqslant{f}\left(\mathrm{2}{x}\right)\leqslant{f}\left(\mathrm{3}{x}\right) \\ $$$$\mathrm{1}\leqslant\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}\leqslant\frac{{f}\left(\mathrm{3}{x}\right)}{{f}\left({x}\right)} \\ $$$${as}\:{x}\Rightarrow\infty\:,\:\mathrm{1}\leqslant\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}\leqslant\mathrm{1} \\ $$$${as}\:{x}\Rightarrow\infty,\:\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}\Rightarrow\mathrm{1} \\ $$

Commented by liberty last updated on 10/Sep/21

thank you

$${thank}\:{you} \\ $$

Answered by puissant last updated on 10/Sep/21

∀x∈R,  x≤2x≤3x ⇒ f(x)≤f(2x)≤f(3x)  (because f ↗).  ⇒ 1≤((f(2x))/(f(x)))≤((f(3x))/(f(x)))  (f(x)≠0)..  ⇒ 1≤ lim_(x→∞) ((f(2x))/(f(x)))≤lim_(x→∞) ((f(3x))/(f(x)))=1  (Gendarm theorem)..  ⇒ lim_(x→∞) ((f(2x))/(f(x)))=1  Answer :  C)

$$\forall{x}\in\mathbb{R}, \\ $$$${x}\leqslant\mathrm{2}{x}\leqslant\mathrm{3}{x}\:\Rightarrow\:{f}\left({x}\right)\leqslant{f}\left(\mathrm{2}{x}\right)\leqslant{f}\left(\mathrm{3}{x}\right) \\ $$$$\left({because}\:{f}\:\nearrow\right). \\ $$$$\Rightarrow\:\mathrm{1}\leqslant\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}\leqslant\frac{{f}\left(\mathrm{3}{x}\right)}{{f}\left({x}\right)}\:\:\left({f}\left({x}\right)\neq\mathrm{0}\right).. \\ $$$$\Rightarrow\:\mathrm{1}\leqslant\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}\leqslant\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{3}{x}\right)}{{f}\left({x}\right)}=\mathrm{1} \\ $$$$\left({Gendarm}\:{theorem}\right).. \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{f}\left(\mathrm{2}{x}\right)}{{f}\left({x}\right)}=\mathrm{1} \\ $$$$\left.{Answer}\::\:\:{C}\right) \\ $$

Commented by liberty last updated on 10/Sep/21

Gendarm theorem = Sequeeze theorem?

$${Gendarm}\:{theorem}\:=\:{Sequeeze}\:{theorem}? \\ $$

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