Find the limit of fraction involving logarithms I am looking for a way to prove the following limit for integer $x$s:
$$\lim_{x\to\infty}{\frac{\log(x+2)-\log(x+1)}{\log(x+2)-\log(x)}}=\frac{1}{2}$$
I could find the result by using a computer program but I cannot formally establish the above equality.

We will use the fundamental limit $$\lim_{x \to 0}\frac{\log(1 + x)}{x} = 1\tag{1}$$ We have
\begin{align}
L &= \lim_{x \to \infty}\frac{\log(x + 2) - \log(x + 1)}{\log(x + 2) - \log x}\notag\\
&= \lim_{x \to \infty}\dfrac{\log\left(\dfrac{x + 2}{x + 1}\right)}{\log\left(\dfrac{x + 2}{x}\right)}\notag\\
&= \lim_{x \to \infty}\dfrac{\log\left(1 + \dfrac{1}{x + 1}\right)}{\log\left(1 + \dfrac{2}{x}\right)}\notag\\
&= \lim_{x \to \infty}\dfrac{\log\left(1 + \dfrac{1}{x + 1}\right)}{\dfrac{1}{x + 1}}\cdot\dfrac{\dfrac{1}{x + 1}}{\dfrac{2}{x}}\cdot\dfrac{\dfrac{2}{x}}{\log\left(1 + \dfrac{2}{x}\right)}\notag\\
&= \frac{1}{2}\lim_{y \to 0}\frac{\log(1 + y)}{y}\cdot\lim_{x \to \infty}\frac{x}{x + 1}\cdot\lim_{z \to 0}\frac{z}{\log(1 + z)}\text{ (putting }y = 1/(x + 1), z = 2/x)\notag\\
&= \frac{1}{2}\notag
\end{align}
