Q. Let $f:R \rightarrow R$ be appositive increasing function with$\lim_{x \rightarrow \infty}{\frac{f(3x)}{f(x)} = 1}$. Then, $\lim_{x \rightarrow \infty}\frac{f(2x)}{f(x)}$ is equal to
A) 1
B) $\frac{2}{3}$
C) $\frac{3}{2}$
D) 3
Explanation
(a)
$f(x)$ is a positive increasing function
$\Rightarrow \ \ \ \ 0 < f(x) < f(2x) < f(3x)$
$\Rightarrow \ \ \ \ 0 < 1 < \frac{f(2x)}{f(x)} < \frac{f(3x)}{f(x)}$
$\Rightarrow \ \ \ \ \ \ \ \lim_{x \rightarrow \infty}1 < \lim_{x \rightarrow \infty}\ \frac{f(2x)}{f(x)} < \lim_{x \rightarrow \infty}\frac{f(3x)}{f(x)}$
By Sandwich theorem
$\lim_{x \rightarrow \infty}\frac{f(2x)}{f(x)} = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.