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Finding a limit of a recurrently given sequence

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I am struggling with one example of limit required for recurrently given sequence. There are 3 examples in total, I will also post the ones I did.

$ x_1>0$ and recursively given sequence: $$x_{n+1}=\frac{x_n}{1+x_n+x_n^2}$$

Find the following limits: $\displaystyle \lim_{n \to \infty }x_n;$$ \displaystyle \lim_{n \to \infty }nx_n;$ and $\displaystyle \lim_{n \to \infty }\frac{n(1-nx_n)}{\log_{e}n}$

First thing I do is determining whether or not the sequence is monotonically increasing or decreasing. (I already know the limit of this sequence will be $\geq 0$ because of $x_1>0$)

$\frac{x_{n+1}}{x_n}$ gives me $\frac{1}{1+x_n+x_n^2}$ which is $<1$ so the sequence monotonically decreases

I take some $L= \displaystyle \lim_{n \to \infty }x_n = \displaystyle \lim_{n \to \infty }x_{n+1}$.Swapping this into the starting sequence I get $L=\frac{L}{1+L+L^2}$ and from here I get $L=0 \vee L=-1$

Because of $x_1>0$, I already know $\displaystyle \lim_{n \to \infty }x_n \neq -1$, so $\displaystyle \lim_{n \to \infty }x_n = 0$ is the only solution here.

The next example is also pretty straight forward, I take $x_n=\frac{y_n}{n}$ and $x_{n+1}=\frac{y_{n+1}}{n+1}$. Here, the result is $\displaystyle \lim_{n \to \infty }nx_n=1$.

The third example, $\displaystyle \lim_{n \to \infty }\frac{n(1-nx_n)}{\log_{e}n}$, looks the trickiest and I couldn't solve it using the method I used in the 2 examples above. If anybody has any idea about solving it please share it. By the way, I am not sure if my method of solving is actually good, so if it's not please recommend/teach me a more versatile method.


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