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Let $x_0 = 1, x_1 = \frac{1}{1+x_0}, x_2 = \frac{1}{1+x_1} \cdots x_n = \frac{1}{1+x_{n-1}}$

Find $\lim_{n\rightarrow \infty} x_n$ as $n$ approaches infinity.

I don't know to represent this recurrence relation in a way that I can solve for $x_n$ and take the limit. Is there another way that I can do this?

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    Have you at least run a few dozen terms to see what is happening? You might find the number looks familiar. What have you tried? Do you know that if there is a limit it must be a fixed point of the recurrence? – 2017-01-23
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    Prove it converges. Then solve$$x=\frac1{1+x}$$and choose the solution through inequalities. – 2017-01-23

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Hint: $x_0=1, x_1=\frac{1}{2}, x_2=\frac{2}{3},x_3=\frac{3}{5},x_4=\frac{5}{8},...$ use induction to conclude that $x_{n}=\frac{F_{n+1}}{F_{n+2}}$, where $F_n$-are Fibonacci numbers. Afterwards use this.