The prime number theorem (statement)

Theorem. $\pi(x)\sim x/\ln x$ as $x\to+\infty$, i.e. $\lim_{x\to+\infty}\pi(x)\ln x/x=1$.

Prediction. For $x=10^6$: $x/\ln x=10^6/\ln10^6=10^6/(6\ln10)\approx10^6/13.816\approx72{,}382$.

Actual value. $\pi(10^6)=78{,}498$, so the prediction is close ($\sim8\%$ low, as expected since the next term is $x/(\ln x)^2$).

Conclusion. The prime number theorem gives the correct leading asymptotic of $\pi(x)$.