Exploring the Limit of f(x)^n: A Homework Challenge

[jobsism]

Homework Statement

Imagine a function: f(x) = x + sin(x)/K where K is pi cubed.
Suppose f(x)^n means f(f(f(f(...f(x))))) n times.
Find the value of lim n->infinity f(x)^n

Homework Equations

The Attempt at a Solution

I have no idea on where to start. Can anyone give a hint, please?

 

[Curious3141]

Homework Statement

Imagine a function: f(x) = x + sin(x)/K where K is pi cubed.
Suppose f(x)^n means f(f(f(f(...f(x))))) n times.
Find the value of lim n->infinity f(x)^n

Homework Equations

The Attempt at a Solution

I have no idea on where to start. Can anyone give a hint, please?

Let the limit be y. Can you see that $f(y) = y$? If the limit exists, then the sequence will converge to that value. Applying the function one more time to the limit will not change the value.

BTW, there's ambiguity in your question. Did you mean $f(x) = x + \frac{\sin x}{K}$ or $f(x) = \frac{x + \sin x}{K}$?

 

[jobsism]

Let the limit be y. Can you see that $f(y) = y$? If the limit exists, then the sequence will converge to that value. Applying the function one more time to the limit will not change the value.

Yes, I understand that much.

BTW, there's ambiguity in your question. Did you mean $f(x) = x + \frac{\sin x}{K}$ or $f(x) = \frac{x + \sin x}{K}$?

Oh, sorry! I meant the former.

 

[Curious3141]

Yes, I understand that much.Oh, sorry! I meant the former.

This is actually a fairly tricky problem. The answer is that there's no single limit. There are, however, an infinite number of "attractors" to which the infinitely iterated function will converge.

Why don't you start by applying the hint in my first post, then we'll take it from there.

Another very important hint: sketch the curve of $y = \sin{\frac{x}{\pi^3}}$.