Let $\displaystyle r(x) = \frac{\pi (x)}{x / \log x}$. Show that for no $\delta >0$ is there a $T= T(\delta )$ such that $r(x) > 1+ \delta$ for all $x>T$, nor is there a $T$ such that $r(x) < 1 - \delta$ for all $x>T$.
I'm stumped on how to approach this. I proved that $$\sum _{p \leq x} \frac{\log p}{p} = \log x + O\left( \frac{\log x}{\log \log x} \right) $$ Is this the right direction? How should I proceed/conclude?
It is a straightforward consequence of the fact that $r(x)\to 1$ as $x\to \infty$.