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

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SUMMARY

The limit of x^x as x approaches 0 is 1, which can be demonstrated using logarithmic properties and L'Hôpital's Rule. By transforming the expression into a product, specifically lim x log x, one can apply L'Hôpital's Rule effectively. Additionally, substituting x with 1/y simplifies the limit to a more recognizable form, aiding in the evaluation process.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with logarithmic functions
  • Knowledge of L'Hôpital's Rule
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the application of L'Hôpital's Rule in various limit problems
  • Explore the properties of logarithms in calculus
  • Learn about limit transformations using variable substitutions
  • Investigate the behavior of exponential functions near zero
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Students and educators in calculus, mathematicians exploring limits, and anyone interested in advanced mathematical analysis techniques.

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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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