How do I show that x^x->1 as x->0?

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Homework Help Overview

The discussion revolves around demonstrating the limit of the expression x^x as x approaches 0, specifically showing that it approaches 1. The subject area involves limits and logarithmic properties in calculus.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants suggest using logarithms to transform the expression and explore the limit. Some express difficulty in establishing a lower bound for the term x log(x). Others mention applying L'Hôpital's rule to analyze the limit further. A participant proposes a substitution to simplify the limit evaluation.

Discussion Status

The discussion includes various approaches to the problem, with participants offering different methods and suggestions. There is no explicit consensus, but several productive directions have been identified, including the use of logarithmic transformations and limit properties.

Contextual Notes

Participants are navigating the complexities of limits involving logarithmic functions and are considering different mathematical techniques to address the problem. The discussion reflects the challenges of working with expressions as they approach critical points.

Treadstone 71
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How do I show that x^x->1 as x->0?
 
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Try using log's to convert the exponentiation into a product.
 
I can't find a lower bound for xlog(x).
 
use l'hopital's rule.

if you have lim x log x then x log x is the same as [log x / (1 / x)]
 
Got it. Thanks.
 
You can also change it to what may be a more familiar limit by setting x=1/y.
 

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