The PNT states that $$\pi(x) \sim \cfrac{x}{\log x} \qquad (x\rightarrow\infty).$$ Let's define a function $M$ to be $$M(x) := \cfrac{\pi(x)-x/\log x}{\lvert\pi(x)-x/\log x\rvert},$$ which returns either $1$ or $-1$, depending on what function $\pi(x)$ or $x/\log x$ is bigger.
In this picture we can clearly see that $\pi(x)$ seems to be much larger than $x/\log x$, where $x$ is approximately greater than $50$. The graph gets bigger and bigger but I think I have heard of a number, let's call it $\Xi$ that satisfies $M(\Xi) = -1$. And not only that - I think some author stated that for $x\in\mathbb{R}, M(x)$ is infinitely many times $-1$, $1$.
1) What is the number $\xi$ called?
2) How can we proof that $M(x)$ changes sign infinitely many often?
(In this post, I will commit the usual abuse of notation/terminology and use $f(x)$ for both the function $f$ and the value of that function at $x$.)
For $x\ge 17$ the graph of $\pi(x)$ is always above the graph of $\frac{x}{\log x}$. See comments to the question for details.
Last last integer $x$ for which $\pi(x) < \frac{x}{\log x}$, is $x=10$. The last real numbers for which this inequality holds, are the open interval $16.99888735\ldots < x < 17$. If you plot $\pi(x)$ with horizontal segments only (no vertical segments at the jumps), the last intersection between the two graphs is therefore $16.99888735\ldots$.
As mentioned also in the comments to the question, if you consider $\mathrm{Li}(x)=\int_2^x \frac1t \mathrm{d}t$ instead of $\frac{x}{\log x}$, then it is true that the graphs of $\pi(x)$ and $\mathrm{Li}(x)$ cross infinitely often (for history, read about Skewes's number).