The graph of the function $f(x)=x+\frac{1}{8} \sin (2 \pi x), 0 \leq x \leq 1$ is shown below. Define $f_1(x)=f(x), f_{n+1}(x)=f\left(f_n(x)\right)$, for $n \geq 1$.
Which of the following statements are true?
$I.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=0$
$II.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=\frac{1}{2}$
$III.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=1$
$IV.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)$ does not exist.
The graph of the function $f(x)=x+\frac{1}{8} \sin (2 \pi x), 0 \leq x \leq 1$ is shown below. Define $f_1(x)=f(x), f_{n+1}(x)=f\left(f_n(x)\right)$, for $n \geq 1$.
Which of the following statements are true?
$I.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=0$
$II.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=\frac{1}{2}$
$III.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=1$
$IV.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)$ does not exist.
- [KVPY 2016]
- A
$I$ and $III$ only
- B
$II$ only
- C
$I,II,III$ only
- D
$I, II, III$ and $IV$
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