Suppose $a\in \mathbb{R}$, $a \in (0,1)$ and a function $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying the following property:
$(1)$ $\lim_{x \rightarrow \infty}{f(x)}=0$
$(2) \lim_{x \rightarrow \infty}{\frac{f(x)-f(ax)}{x}}=0$
Show that $\lim_{x \rightarrow \infty}{\frac{f(x)}{x}}=0$
This is a problem from 'Putnam and Beyond', page $127$.
My attempt: Clearly $\lim_{x \rightarrow \infty}{f(x)}=0 \Rightarrow \lim_{x \rightarrow \infty}{f(ax)}=0 \Rightarrow \lim_{x \rightarrow \infty}{\frac{f(ax)}{x}}=0 \Rightarrow \lim_{x \rightarrow \infty}{\frac{f(x)}{x}}=0$.
Question: Does the last arrow hold? I obtain it using $(2)$ and distributivity of limit. But I don't know whether the limit $\lim_{x \rightarrow \infty}{\frac{f(x)}{x}}$ exists or not.