A 'rational function' $\displaystyle \rho (x)$ can be written as...
$\displaystyle \rho(x)= a \cdot \frac{N(x)}{D(x)}$ (1)
we don't need that $\displaystyle a$.
... where $\displaystyle N(x)$ and $\displaystyle D(x)$ are polynomial in x of degree n and m so that we can write...
$\displaystyle \rho(x) \sim k\cdot x^{n-m}$ (2)
... or equivalently...
$\displaystyle \lim_{x \rightarrow \infty} \frac{\rho (x)}{k \cdot x^{n-m}} =1$ (3)
At the end of nineteenth century it has been demonstrated that is...
$\displaystyle \pi(x) \sim \frac{x}{\ln x}$ (4)
... or equivalently...
$\displaystyle \lim_{x \rightarrow \infty} \frac{\pi (x)}{\frac{x}{\ln x}} =1$ (5)
Combining (3) and (5) we find that is...
$\displaystyle \lim_{x \rightarrow \infty} \frac{\rho(x)}{\pi(x)}= \left\{\begin {array}{cc}\infty,&\mbox{if }n-m>0\\0,&\mbox{if }n-m \le 0\end{array}\right.$ (6)
... so that the situation described by NCA is impossible...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$