<s>[INST] Proving $\lim_{x \to +\infty}\frac{\ln(x)}{2\cdot \ln(x+1)} = \frac{1}{2}$ with $\epsilon- \delta$ Today, I spent some hours trying to solve this problem but I can't reach the conclusion:

Show with $\epsilon-\delta $ definition the following limit:
$$\lim_{x \to +\infty}\frac{\ln(x)}{2\cdot \ln(x+1)} = \frac{1}{2}$$

So, first of all, I set up the inequality:
$$\left|\frac{\ln(x)}{2\cdot \ln(x+1)}-\frac{1}{2}\right|<\epsilon$$
Then:
$$\frac{1}{2}-\epsilon<\frac{\ln(x)}{2\cdot \ln(x+1)}<\frac{1}{2}+\epsilon \implies (1-2\epsilon)\cdot \ln(x+1) < \ln(x) < (1+2\epsilon)\cdot \ln(x+1)$$
And elevating with base $e$ to cancel the $\ln(x)$:
$$(x+1)^{1-2\epsilon}<x<(x+1)^{1+2\epsilon}$$
I want to apply the limit $\lim_{x \to +\infty}\frac{(1+f(x))^\alpha-1}{\alpha\cdot f(x)}=1$ with $f(x) \to_{x \to 0} 0$. So, I have:
$$x^{1-2\epsilon}\cdot\left(1+\frac{1}{x}\right)^{1-2\epsilon}<x<x^{1+2\epsilon}\cdot\left(1+\frac{1}{x}\right)^{1+2\epsilon}$$
Then:
$$
\left\{\begin{matrix}
x>x^{1-2\epsilon}\cdot\left(1+\frac{1}{x}\right)^{1-2\epsilon}
\\ x<x^{1+2\epsilon}\cdot\left(1+\frac{1}{x} \right )^{1+2\epsilon}
\end{matrix}\right.
$$
So:
$$
\left\{\begin{matrix}
x^{2\epsilon} - 1>\left(1+\frac{1}{x}\right)^{1-2\epsilon} - 1
\\ \frac{1}{x^{2\epsilon}}-1<\left(1+\frac{1}{x} \right )^{1+2\epsilon}-1
\end{matrix}\right.
$$
When $x$ approaches $+\infty$:
$$
\sim
\left\{\begin{matrix}
x^{2\epsilon}-1>\frac{1-2\epsilon}{x}
\\\frac{1}{x^{2\epsilon}}-1<\frac{1+2\epsilon}{x}
\end{matrix}\right.
$$
Notice that the second inequality is always true because $\frac{1}{x^{2\epsilon}}\to 0^+$ when $x \to +\infty$, so that left side is negative while right side $\frac{1-2\epsilon}{x}$ is positive.
Now, in some way I have to find $\delta = N$. I noticed that: $$x^{2\epsilon}-1\sim x^{2\epsilon} \;\;\; x \to +\infty$$
and then for the first inequality:
$$x^{2\epsilon}>\frac{1-2\epsilon}{x} \implies x> \delta(\epsilon)= N =(1-2\epsilon)^{1+2\epsilon}$$
Here, tere is a problem because when $\epsilon \to 0^+$, $\delta = N \to 1$ and not to $+\infty$. Is this anyway correct? Or, I can't apply asympothics reduction with $\epsilon - \delta$?
Edit:
I  all three solutions proposed, $\delta = N$ goes to infinity when $\epsilon \to 0^+$. In my answer, $N$ tends to $1$, so is my solution not corret? If it's yes, why?
Thanks.
 [/INST] Fix $x>e^2-1$. Using
$$ \ln(1+\frac1x)<\frac1x $$
we have
$$ \left|\frac{\ln(x)}{2\cdot \ln(x+1)}-\frac{1}{2}\right|=\frac12\frac{\ln(1+\frac1x)}{\ln(x+1)}<\frac{1}{4x}. $$
Then for $\forall \varepsilon>0$, denote $N$ by $N=\max\{e^2-1,\frac1{4\varepsilon}\}$. Then when $x>N$,
$$ \left|\frac{\ln(x)}{2\cdot \ln(x+1)}-\frac{1}{2}\right|<\varepsilon. $$
</s>