Example of Function f:R-R where lim x→0 xf(1/x) ≠ 0

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Discussion Overview

The discussion revolves around finding an example of a function \( f: \mathbb{R} \to \mathbb{R} \) such that the limit \( \lim_{x \to 0} x f(1/x) \) is not equal to zero. The scope includes mathematical reasoning and exploration of function properties.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant requests assistance in finding a function that meets the specified limit condition.
  • Another participant suggests that if \( f(1/x) \) can cancel the \( x \), then the limit could be independent of \( x \).
  • A participant proposes that \( f(x) \) might need to be \( -x^2 \) if \( f(1/x) \) is to be proportional to \( 1/x \).
  • A similar suggestion is reiterated, emphasizing the relationship between \( f(1/x) \) and \( 1/x \), with a hint to relabel variables for clarity.

Areas of Agreement / Disagreement

Participants express various hypotheses about the form of \( f(x) \) but do not reach a consensus on a specific function that satisfies the limit condition.

Contextual Notes

The discussion includes assumptions about the behavior of the function \( f \) and its relationship to \( 1/x \), but these assumptions are not fully explored or resolved.

Trigger
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Please help with the following problem:

Find an example of a finction f:R-R where

lim x tends towards zero of xf(1/x) is not zero
 
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It would obviously be the case when f(1/x) can cancel the x, so that the limit is independent of x.
What would f(x) have to be, if f(1/x) is to be proportional to 1/x?
 
Would f(x) have to be -x squared if f(1/x) is to be proportional to 1/x
 
Trigger said:
Would f(x) have to be -x squared if f(1/x) is to be proportional to 1/x

Hint:Relabel 1/x with "lambda",get the old function f(1/x) in terms of "lambda" and then simply rename "lambda" as being x.U'll get your answer.
 

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