Proving Convergence of x^x to 1 at x->0

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The discussion focuses on proving that the limit of x^x converges to 1 as x approaches 0. The user successfully transforms x^x into e^(x ln(x)), which allows for the application of L'Hôpital's Rule to demonstrate that the limit of x ln(x) approaches 0. However, the user encounters difficulties in proving this convergence using the Epsilon-Delta definition of limits. The conversation highlights the challenges of defining limits for functions that are undefined at certain points.

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I am trying to prove that the upper +limit of x^x, when x->0 converges to 1.

So I started by converting x^x to e^(x ln(x)). I know that this eliminates the domain: x <= 0, but I still believe that I can still continue on.

So here I tried to constrain the limit: x ln(x) (i.e. x->0, x ln(x) -> 0; which is where I failed. Although I can show via the L'hopital's Rule that it is true, I struggle to show it via the Epsilon Delta Definition.

I know that x^x is undefined at 0, but I still want to show that the curve converges towards 1.
 
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Why not show that L'Hopitul's rule can be proven in general with the epsilon/delta-definition?
 

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