What Is the Limit of x^x as x Approaches 0?

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Discussion Overview

The discussion revolves around the limit of the expression x^x as x approaches 0, exploring both the limit's value and the implications of the expression 0^0.

Discussion Character

  • Exploratory, Technical explanation, Homework-related

Main Points Raised

  • One participant asserts that the limit of x^x as x approaches 0 is 1 but expresses difficulty in proving it.
  • Another participant suggests considering the value of x to the 0 power as a hint for understanding the limit.
  • A different participant proposes a mathematical transformation involving the exponential and logarithmic functions to analyze the limit.
  • One participant agrees that the limit of x^x is 1 as x approaches 0 but emphasizes that this does not resolve the question of the value of 0^0 or its uniqueness in limits.
  • A participant shares an image link, possibly as a visual aid related to the discussion.
  • Another participant expresses disappointment in their own creative thinking regarding the problem.
  • A first-year electrical engineering student reassures the disappointed participant that the problem is trivial and indicates that they are accustomed to more complex topics.

Areas of Agreement / Disagreement

There is a general agreement that the limit of x^x as x approaches 0 is 1, but there is no consensus on the implications regarding 0^0, which remains a point of contention.

Contextual Notes

Participants note that the limit does not imply a definitive value for 0^0, and the uniqueness of this limit is not established.

marshmellow
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Though I know that the limit as x approaches 0 of x^x is 1, I can't prove it...

...can anyone please help me?
 
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hint: ask yourself what x to the 0 power is.
 
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hint write
x^x=exp(log(x)/(1/x))
 
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Hello marshmellow
certainly you will succeed in proving that the limit for x^x is 1 while x tends to 0.
But remember, this doesn't prove that 0^0 = 1 and most certainly not the uniqueness of 0^0 as limit of x^y while both x and y tend to 0.
 
http://pokit.etf.ba/get/e57018aced28181afefff3a8e5a3e402.jpg

there you go, njoy
 
Last edited by a moderator:
thank you very much, I'm actually quite disappointed i can't think that creatively
 
I am first year electrical engineering, this is trivial for me, I have to know much more complicated things (: so don't be disappointed
 

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