Proving Limit at Infinity: n^(1/n) = 1

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Homework Help Overview

The discussion revolves around proving the limit of the expression n^(1/n) as n approaches infinity, specifically addressing the question of whether this limit equals 1. The original poster raises a concern about the indeterminate form of infinity to the power of zero.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the nature of indeterminate forms and suggest taking the logarithm of the expression to analyze the limit. There is also a mention of L'Hopital's rule as a potential method for evaluation.

Discussion Status

The conversation is ongoing, with participants exploring different approaches to the problem. Some guidance has been offered regarding the use of logarithms and L'Hopital's rule, but no consensus has been reached on the best method to prove the limit.

Contextual Notes

Participants are considering the implications of indeterminate forms in the context of limits and are questioning the assumptions underlying the limit evaluation.

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Homework Statement



How can I prove that:

[tex]\lim_{n \rightarrow \infty} n^{\frac{1}{n}}=1[/tex]

Isn't [tex]\infty^{0}[/tex] indeterminate?
Thanks!


Homework Equations





The Attempt at a Solution

 
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Yes, it is indeterminant. That's not the end of the story. Indeterminant just means you don't know what the limit is yet. Take the log. Can you prove (1/n)*log(n) approaches 0?
 
That becomes [tex]0*\infty[/tex], isn't that indeterminate as well?
 
Can't you use L'Hopital's rule?
 
Right. Thanks guys!
 

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