Mathematics - Limits Question with Solution | TestHub

$f(x)$ is a positive increasing function $$\begin{array}{rl}& \Rightarrow 0<f(x)<f(2x)<f(3x)\\ & \Rightarrow 0<1<\frac{f(2x)}{f(x)}<\frac{f(3x)}{f(x)}\\ & \Rightarrow \underset{x\to \infty }{\lim }1\le \underset{x\to \infty }{\lim }\frac{f(2x)}{f(x)}\le \underset{x\to \infty }{\lim }\frac{f(3x)}{f(x)}\end{array}$$ By sandwich theorem. $$\Rightarrow \underset{x\to \infty }{\lim }\frac{f(2x)}{f(x)}=1$$