Let f: R (0,) be strictly increasing function such that x lim(f(7x))/(f(x))=1.

Question

Let $f:	o R	o (0,\infty)$ be strictly increasing function such that $\underset{x	o \infty}{lim}\frac{f(7x)}{f(x)}=1$. Then, the value of $\underset{x	o \infty}{lim}[\frac{f(5x)}{f(x)}-1]$ is equal to

Accepted Answer

$\begin{gathered} f:R	o (0,\infty) \ \underset{x	o \infty}{lim}\frac{f(7x)}{f(x)}=1 \end{gathered}$

$\because f$ is increasing

$\begin{gathered} 	herefore f(x)<f(5x)<f(7x) \ \because \frac{f(x)}{f(x)}<\frac{f(5x)}{f(x)}<\frac{f(7x)}{f(x)} \ 1<\underset{x	o \infty}{lim}\frac{f(5x)}{f(x)}<1 \ 	herefore [\frac{f(5x)}{f(x)}-1] \ \Rightarrow 1-1=0 \end{gathered}$

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