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Calculus Level 2

$\large \lim_{n\to\infty} \underbrace{\sin( \sin ( \sin (\cdots ( \sin(x)))\cdots )}_{n \text{ number of sines}}$

Compute the limit above, where $x$ is any real number .

\frac12

0

\frac{\sqrt3}2

\frac1{\sqrt2}

3 solutions

Sabhrant Sachan
May 29, 2016

$f(x)=\sin{x} \in [-1,1] \\ f(x)=\sin{\left(\sin{x}\right)} \in [\sin{(-1)},\sin{(1)}] \\ f(x)=\sin{\left(\sin{\left(\sin{x}\right)}\right)} \in [\sin{\left(\sin{(-1)}\right)},\sin{\left(\sin{(1)}\right)}]$

As you can see,the Range of the function is deceasing , when the $\displaystyle\lim_{n \to \infty}$ , Range $(R) \rightarrow 0$ , Look at this graph for better understanding.