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

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In summary, the conversation discusses the limit of x^x as x approaches 0 and how it can be proven to be 1. It also mentions the fact that this does not prove the uniqueness of 0^0 as the limit of x^y while both x and y tend to 0. The conversation ends with someone providing a helpful hint and offering encouragement to the original poster, who is struggling with the concept.
  • #1
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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  • #2
hint: ask yourself what x to the 0 power is.
 
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Likes Stormy Sea
  • #3
hint write
x^x=exp(log(x)/(1/x))
 
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Likes 101pritam
  • #4
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.
 
  • #5
http://pokit.etf.ba/get/e57018aced28181afefff3a8e5a3e402.jpg

there you go, njoy
 
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  • #6
thank you very much, I'm actually quite disappointed i can't think that creatively
 
  • #7
I am first year electrical engineering, this is trivial for me, I have to know much more complicated things (: so don't be disappointed
 

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

What is the limit of x^x as x approaches 0?

The limit of x^x as x approaches 0 is equal to 1. This can be determined by using L'Hopital's rule or by taking the limit as x approaches 0 from both the left and right sides of the function.

Why is the limit of x^x as x approaches 0 equal to 1?

The limit of x^x as x approaches 0 is equal to 1 because as x gets closer and closer to 0, the value of x^x approaches 1. This can be seen by graphing the function or by using small values of x to evaluate the limit.

Is the limit of x^x as x approaches 0 undefined?

No, the limit of x^x as x approaches 0 is not undefined. It is equal to 1, as explained by the definition of a limit and the behavior of the function as x approaches 0.

Can the limit of x^x as x approaches 0 be calculated using algebra?

No, the limit of x^x as x approaches 0 cannot be calculated using algebra. It requires the use of calculus, specifically L'Hopital's rule, to determine the limit.

What is the significance of the limit of x^x as x approaches 0?

The limit of x^x as x approaches 0 has significance in the study of calculus and the behavior of functions. It also has applications in areas such as economics and biology.

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