If $x_0=1$ and $x_n=\frac {1}{1+x_{(n-1)}}$, find: $\lim_{x\to\infty} x_n$ 
If $x_0=1$ and $x_n=\dfrac {1}{1+x_{(n-1)}}$, find $\displaystyle\lim_{x\to\infty} x_n$.

My attempt:
$x_1=1+\dfrac 1 2=\dfrac 3 2$
$x_2=1+\dfrac {1}{1+\frac 3 2}=\dfrac2 5$
Which gives following series:
$$1, \frac32, \frac35,\frac 58, \frac 8 {13}, \dots$$
Since the denominator is greater than the numerator, the limit of $x_n=0$ when $x\to\infty$. I do not think this result is correct. I think this problem must have a solution using continued fractions. Any idea?

First, try to show that a limit even exists in the first place. Then, in doing so, let
$$L := \lim_{n \to \infty} x_n$$
With this, then, taking the limit of both sides of your equation gives us
$$L = \frac{1}{1+L}$$
This will give you two possible values for $L$. Use $x_0$ and a convergence argument to determine which $L$ is the value relevant for your case.

You are right in that this references continued fractions in a sense. If you imagine expanding the recursion backwards, you'll get
$$L = \cfrac{1}{1+ \cfrac{1}{1+ \cfrac{1}{1+ \cfrac{1}{1+ \ddots}}}}$$
There is a self-similarity here: namely everything below the second numerator is in fact just $L$ itself:
$$L = \frac{1}{1+L}$$
However, this is not a fully rigorous way to handle the problem. For instance, it tells you possible values of $L$, but not which ones actually are valid - or, indeed, if any of them are. (Heck, the value of $x_0$ is important, even: there's a value for $x_0$ in this case in which you would get a different $L$ than in your case and in nearly every other case.) An infinite process like this is almost always defined in terms of a limit of the smaller, finite process. For instance, like how we define
$$\sum_{k=1}^\infty a_n := \lim_{n \to \infty} \sum_{n=1}^k a_n$$
We'd first have to know that the limit in question even exists in the first place before assigning the infinite sum a value via any other means. Such is mathematical rigor.
I give a further elaboration on this sort of notion in this answer which ties into continued fractions (particularly the well-known one for $\varphi = (1 + \sqrt 5)/2$ and its conjugate), how we define these, and the convergence of a continued fraction's "partial fractions."
