# Gauss Limit using Prime Number Theorem

I need to show $\lim_{n\to\infty}\frac{Li(n)}{\pi(n)} =1$, where $Li(n)$ = $\int_{2}^{n} \frac{dx}{ln(x)}$.

I know I need to use the Prime Number Theorem, and L'Hopital's rule. However, I am I can't seem to get it started correctly. Can you help?

Recall that $$\frac{d}{dx} \frac{x}{\log x} = \frac{\log x -1}{(\log x)^2}$$ so that $$\lim_{x \to \infty} \frac{Li(x)}{\pi (x)} = \lim_{x \to \infty} \frac{Li(x)}{x/ \log x} = \lim_{x \to \infty} \frac{1/\log x}{(\log x -1)/ (\log x)^2} =1$$ where the first equality follows by the PNT, and the second equality follows by applying L'Hopital.