calculus - Limit (without series expansion and l'Hôpital's rule)

$$\lim_{x \to \infty}\ln{\frac{x+\sqrt{x^2+1}}{x+\sqrt{x^2-1}}}\cdot \left(\ln{\frac{x+1}{x-1}}\right)^{-2}=\frac{1}{8}$$

Any suggestion to find this limit without series expansion and l'Hôpital's rule? Thanks and regards.

Note:
WolframAlpha confirms that the result is $\frac{1}{8}$.

Answer

It is easy to observe that as $x \to \infty$ the argument of both the logarithms tend to $1$ and this is exactly what we need when we are dealing with limit of logarithmic terms. Just add $1$ and subtract $1$ in the argument of log and proceed as follows:
$$\begin{align} L &= \lim_{x \to \infty}\log\frac{x + \sqrt{x^{2} + 1}}{x + \sqrt{x^{2} - 1}}\left(\log\frac{x + 1}{x - 1}\right)^{-2}\notag\\ &= \lim_{t \to 0^{+}}\log\frac{1 + \sqrt{1 + t^{2}}}{1 + \sqrt{1 - t^{2}}}\left(\log \frac{1 + t}{1 - t}\right)^{-2}\text{ (putting }x = 1/t)\notag\\ &= \lim_{t \to 0^{+}}\log\left(1 + \frac{1 + \sqrt{1 + t^{2}}}{1 + \sqrt{1 - t^{2}}} - 1\right)\left\{\log\left(1 + \frac{1 + t}{1 - t} - 1\right)\right\}^{-2}\notag\\ &= \lim_{t \to 0^{+}}\log\left(1 + \frac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{1 + \sqrt{1 - t^{2}}}\right)\left\{\log\left(1 + \frac{2t}{1 - t}\right)\right\}^{-2}\notag\\ &= \lim_{t \to 0^{+}}\dfrac{\log\left(1 + \dfrac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{1 + \sqrt{1 - t^{2}}}\right)}{\dfrac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{1 + \sqrt{1 - t^{2}}}}\cdot\dfrac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{1 + \sqrt{1 - t^{2}}}\left\{\log\left(1 + \frac{2t}{1 - t}\right)\right\}^{-2}\notag\\ &= \lim_{t \to 0^{+}}\dfrac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{1 + \sqrt{1 - t^{2}}}\left\{\dfrac{\log\left(1 + \dfrac{2t}{1 - t}\right)}{\dfrac{2t}{1 - t}}\cdot\dfrac{2t}{1 - t}\right\}^{-2}\notag\\ &= \lim_{t \to 0^{+}}\dfrac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{1 + \sqrt{1 - t^{2}}}\cdot\frac{(1 - t)^{2}}{4t^{2}}\notag\\ &= \frac{1}{8}\lim_{t \to 0^{+}}\frac{\sqrt{1 + t^{2}} - \sqrt{1 - t^{2}}}{t^{2}}\notag\\ &= \frac{1}{8}\lim_{t \to 0^{+}}\frac{2t^{2}}{t^{2}\left\{\sqrt{1 + t^{2}} + \sqrt{1 - t^{2}}\right\}}\notag\\ &= \frac{1}{8}\notag \end{align}$$
The technique used above is based on the standard limits and is well explained in the comments to this answer.