Lower bound for number of primes up to $x$.

Question

It is possible to prove the statement that $\pi(x)$, the number of primes up to $x$ is bigger than $\sqrt x$ for $x \ge 3$, in an elementary way? More generally can we prove that for every $0<\epsilon<1$, $$ x^\epsilon < \pi(x), $$ for $x$ large enough, in an elementary way?