I'm looking at Theorem 3 of this paper, which is

If $\:x\geq \exp(\exp(45))\:$ and $\;\;h\:\geq\:3\cdot x^{\frac23}\;\;$ then $$\pi(x+h)-\pi(x) \; \geq \; h\cdot \left(1-\left(3192.34\cdot \exp\left(\left(-\frac1{283.79}\right)\cdot \left(\left(\frac{\log(x)}{\log(\log(x))}\right)^{\frac13}\right)\right)\right)\right)$$

The term starting with 3192.34 goes to zero as $x$ goes to infinity, which results in

$\pi(x+h)-\pi(x) \: \approx \: h \: = \: (x+h)-x \: \not\approx \frac{(x+h)-x)}{\log(x)} \;\;$,

contradicting the prime number theorem.

What am I missing here?