What is the Limit of x^((x^x)-1) as x Approaches Zero?

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Homework Help Overview

The discussion centers around evaluating the limit of the expression x^((x^x)-1) as x approaches zero, which presents an indeterminate form of 0^0. Participants are exploring methods to analyze this limit within the context of calculus.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the use of logarithmic transformation and L'Hôpital's rule to evaluate the limit, with some questioning the practicality of these methods. There is also a suggestion to express the limit in terms of ln(y) for further analysis.

Discussion Status

The conversation is ongoing, with participants sharing their thoughts on the effectiveness of L'Hôpital's rule and expressing uncertainty about alternative approaches. There is recognition of the complexity involved in solving the limit without this method.

Contextual Notes

Participants note the indeterminate form of the limit and the potential challenges in finding a straightforward solution. The discussion reflects a collaborative effort to navigate the problem's intricacies without arriving at a definitive conclusion.

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Homework Statement


Limit of x^((x^x)-1) as x->0


Homework Equations


Lim x^x=1
x->0


The Attempt at a Solution


it's an 0^0 indetermination so I tried to solve it the usual way, by first calculating the limit of log(x)*(x^x-1) as x->0 with L'hopital's rule. I got e^0=1 confirmed by Wolfram. However, the using l'hospital's rule on this limit is not very practical, is there a better soluction?
 
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Suppose you let y = x^((x^x)-1), what is ln(y) = ?
 
I would get the same limit log(x)*(x^x-1) as x->0 but solving this by L'hopital's rule takes a while or is it the fastest way?
 
Oh whoops, I read too quickly. Yeah I don't see how you would do this without LH otherwise.
 

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