Find the limit of composition 1

Calculus Level 3

A function is defined as $f(x)=\sin x$ and $f \circ f \circ f \circ \cdots \circ f(x)=f_n (x)$ , where $n$ is the number of compositions, for all real $x$ .

Find $\displaystyle \lim_{n \to \infty} f_n (x)$ .

The answer is 0.

1 solution

Hjalmar Orellana Soto
Dec 18, 2016

As we know, using (https://brilliant.org/wiki/taylor-series/)[Taylor Series], we have: $\sin(x)=x$ For every $x\leadsto 0$ . So as we compose the function $f(x)=\sin(x)$ we are changing the domain of the function to $[-1,1]$ which is an interval close to $0$ , and each composition make the function's value approach to $0$ , so composing infinitely the funcion $f$ the function becomes null

Find the limit of composition 1

The answer is 0.

1 solution

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