Possible Duplicate: Explaining $\cos^\infty$

Does the following limit exist?

$$\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$$

If yes, find the limit.

If no, please explain why the limit doesn't exist.

I think, the limit exists.

So, I tried to use Squeeze theorem but didn't work.

  • $\begingroup$ You should guess what the limit should be, then prove that it actually is the limit. $\endgroup$
    – Tunococ
    Jan 28, 2013 at 14:32
  • $\begingroup$ yes, I also try to use a sequence instead the limit. $a_1=\cos x$ and $a_{n+1}=\cos(a_n)$ , but my friend said that was wrong $\endgroup$
    – cwk709394
    Jan 28, 2013 at 14:35
  • 6
    $\begingroup$ See here. $\endgroup$ Jan 28, 2013 at 14:35

1 Answer 1


You may treat is as dynamical system with state transition function $f(x) = cos(x)$. After first two iterations $f^{n > 2}(x)$ will lay in interval $I = [cos(1), 1]$. Line $g(x) = x$ will intercept $cos(x)$ in interval $I$ exactly once so $cos(x)$ has unique fixed point in $I$. Because of unique fixed point and because $|f'(x)| < 1$ sequence $f^n(x)$ will converge to this fixed point.

  • $\begingroup$ Consider adding this answer to the earlier duplicate, linked above. Rather than calling it a "dynamical system", though, it would be a little more self- contained to call it a fixed-point iteration. $\endgroup$
    – hardmath
    Jan 28, 2013 at 15:11
  • 1
    $\begingroup$ I have copied this answer to duplicate post. $\endgroup$ Jan 28, 2013 at 15:18

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