$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $

Question

Let $\ π(x)$ denote the prime counting function , i.e. the number of primes not exceeding $x$
Then does $$ \ \lim_{x\to ∞ }\frac{π(x)} { x^δ} $$ exist for all real $δ$ $∈ ( 0 , 1 )$

Answer 1

No, because the prime number theorem states that $\pi(x) \approx \frac{x}{\log{x}}$. Since $\log{x} \lt x^\delta$ for any $\delta \gt 0$ and sufficiently large $x$, the limit diverges.