Estimates of the bounds of $\pi (x)$ and $\pi ((x+1{)}^2)-\pi (x^2)$

We show the following bounds on the prime counting function $\pi (x)$ using principles from analytic number theory, giving an estimate: $$2\log 2\ge \underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{\pi (x)}{x\mathrm{/}\log x}\ge \underset{x\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{\pi (x)}{x\mathrm{/}\log x}\ge \log 2$$ for all $x$ sufficiently large. We also conjecture about the bounding of $\pi((x+1)^{2}) - \pi(x^{2})$, as is relevant to Legendre’s conjecture about the number of primes in the aforementioned interval such that: $$\lfloor \frac{1}{2}(\frac{{(x+1)}^2}{\log (x+1)}-\frac{x^2}{\log x})-\frac{{(\log x)}^2}{\log (\log x)}\rfloor \le \pi ((x+1{)}^2)-\pi (x^2)\le$$ $$\lfloor \frac{1}{2}(\frac{{(x+1)}^2}{\log (x+1)}-\frac{x^2}{\log x})+{\log }^{2}x\log \log x\rfloor$$