$\lim_{x \to \infty}\ln{\frac{x+\sqrt{x^2+1}}{x+\sqrt{x^2-1}}}\cdot \left(\ln{\frac{x+1}{x-1}}\right)^{-2}=\frac{1}{8}$
Any suggestion to find this limit without series expansion and l'Hôpital's rule? Thanks and regards.
Note: WolframAlpha confirms that the result is $\frac{1}{8}$.