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Let \mathrm{f: R \rightarrow R} be a positive increasing function with \mathrm{ \lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1. Then, \lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=}
 

Option: 1

\frac{3}{2}


Option: 2

3


Option: 3

1


Option: 4

\frac{2}{3}


Answers (1)

best_answer

It is given that \mathrm{f(x)} is a positive increasing function.

\mathrm{ \therefore 0<f(x)<f(2 x)<f(3 x) \\ }

\mathrm{ \Rightarrow 0<1<\frac{f(2 x)}{f(x)}<\frac{f(3 x)}{f(x)} \Rightarrow 1<\frac{f(3 x)}{f(x)} }

Clearly, \mathrm{ \lim _{x \rightarrow \infty} 1=1\: and \: \lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1 }.

Therefore, by Sandwich theorem, we have \mathrm{ \lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=1. }

Hence option 3 is correct.

Posted by

avinash.dongre

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