Why Is 1^∞ an Indeterminate Form in Limits?

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Discussion Overview

The discussion revolves around the concept of why \(1^\infty\) is considered an indeterminate form in the context of limits. Participants explore the implications of approaching the values of 1 and infinity in various functions, examining the nuances involved in limit calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the classification of \(1^\infty\) as indeterminate, suggesting that \(1\) raised to any power should simply equal \(1\).
  • Another participant proposes that perturbing the value of \(1\) can lead to results of either zero or infinity, indicating the sensitivity of limits in this context.
  • Some participants argue that the behavior of the function approaching \(1\) or \(\infty\) is crucial, noting that the speed at which it approaches these values can significantly affect the limit outcome.
  • There is a comparison made to the indeterminate form \(0/0\), where the limits of the numerator and denominator could approach zero at the same rate, leading to different limit results.
  • Several participants emphasize that the function never actually reaches the value \(1^\infty\) and that the analysis focuses on behavior around that point rather than at it.

Areas of Agreement / Disagreement

Participants express a range of views regarding the nature of \(1^\infty\) as an indeterminate form, with no clear consensus reached. The discussion reflects differing interpretations of how limits behave in this context.

Contextual Notes

The discussion highlights the dependence on the speed of approach to the values involved and the importance of considering limits in a neighborhood around a point rather than at the point itself.

kbaumen
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I'm just curious, why, when solving limits, is 1^\infty considered an indeterminate form? Isn't 1 raised to any power equal to 1? Why isn't it so simple?
 
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Probably because if you perturb the 1 at all, the result is either zero or infinite.
 
Well that is a very good question but the problem is that it depends on how fast your function is going to 1 or infinity.

Your function can be going so slowly to 1, in which case the limit goes to 0.

It`s the same as the undetermined form 0/0 the function on top and bottom could approach zero at the same speed and the limit could go to 1.
 
╔(σ_σ)╝ said:
Well that is a very good question but the problem is that it depends on how fast your function is going to 1 or infinity.

Your function can be going so slowly to 1, in which case the limit goes to 0.

It`s the same as the undetermined form 0/0 the function on top and bottom could approach zero at the same speed and the limit could go to 1.

Ah, yes, the function never actually reaches the "value" 1^\infty, since we never consider the function at exatcly x=a, we are just curious about what it does around that point.

Thank you for your answers.
 
Last edited:
kbaumen said:
Ah, yes, the function never actually reaches the value 1^\infty, since we never consider the function at exatcly x=a, we are just curious about what it does around that point.

Thank you for your answers.

Precisely, you are correct. Even in the precise definition of a limit we only look at the deleted neighborhood of x .

You are welcome.:)
 

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