What is the limit of f(x)^n as n approaches infinity?

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Homework Help Overview

The discussion revolves around the limit of a function defined as f(x) = x + sin(x)/K, where K is pi cubed, as it is iteratively applied n times. Participants are exploring the behavior of this function as n approaches infinity.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Some participants question the initial setup of the function and its implications for the limit. Others suggest defining the limit as y and exploring the condition f(y) = y to investigate convergence. There is also a discussion about the ambiguity in the function's definition.

Discussion Status

Participants are actively engaging with the problem, with some offering hints and others seeking clarification. There is recognition of the complexity of the problem, and suggestions to explore graphical representations of the function are made.

Contextual Notes

There is noted ambiguity in the formulation of the function, which may affect the interpretation of the limit. The discussion also hints at the possibility of multiple attractors rather than a single limit.

jobsism
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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?
 
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jobsism said:

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 [itex]f(y) = y[/itex]? 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 [itex]f(x) = x + \frac{\sin x}{K}[/itex] or [itex]f(x) = \frac{x + \sin x}{K}[/itex]?
 
Curious3141 said:
Let the limit be y. Can you see that [itex]f(y) = y[/itex]? 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.

Curious3141 said:
BTW, there's ambiguity in your question. Did you mean [itex]f(x) = x + \frac{\sin x}{K}[/itex] or [itex]f(x) = \frac{x + \sin x}{K}[/itex]?
Oh, sorry! I meant the former.
 
jobsism said:
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 [itex]y = \sin{\frac{x}{\pi^3}}[/itex].
 
Last edited:

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