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Let $f_{1}:[0,4] \rightarrow[0,4]$ be defined by $f_{1}(x)=3-(x / 2)$. Define $f_{n}(x)=$ $f_{1}\left(f_{n-1}(x)\right)$ for $n \geq 2$.

  • Prove that $\displaystyle{}\lim _{n \rightarrow \infty} f_{n}(0)$ exists.
  • Find the set of all $x$ such that $\displaystyle{}\lim _{n \rightarrow \infty} f_{n}(x)$ exists and also find the corresponding limits.
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