Interchanging limits with the prime counting function

Question

How does one justify that $$\lim_{s \to 1} \lim_{x \to \infty} \frac{\pi(x)}{x^s} = \lim_{x \to \infty} \lim_{s \to 1} \frac{\pi(x)}{x^s}, \quad s > 1,$$ without using the fact that the primes have density zero, i.e., $\lim_{x \to \infty} = \pi(x)/x = 0$, and the prime number theorem.