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A nice limit
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mathematica
Poincare Conjecture
Poincare Conjecture


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#1
A nice limit
Proove the following limit: \lim_{x\to 0+}\frac{1}{x^{x}}=1
Last edited by mathematica on Wed Jul 19, 2006 4:07 am; edited 1 time in total 
PostPosted: Sun Sep 18, 2005 12:23 am  Back to top 
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liyi
Navier-Stokes Equations
Navier-Stokes Equations

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#2
Where is x? Do you mean \frac{1}{x^x}\to 1?

\lim_{x\to 0^+} x^x = \lim_{x\to 0^+} e^{x\ln x} = e^{\lim_{x\to 0^+} x\ln x} = e^0 = 1.

PostPosted: Sun Sep 18, 2005 12:38 am  Back to top 
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mathematica
Poincare Conjecture
Poincare Conjecture


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#3
how do you prove \lim_{x\to 0^{+}}e^{x\ln x}= e^{\lim_{x\to 0^{+}}x\ln x}
Last edited by mathematica on Wed Jul 19, 2006 4:07 am; edited 1 time in total 
PostPosted: Mon Sep 19, 2005 4:25 am  Back to top 
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Matnomi
Hodge Conjecture
Hodge Conjecture


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#4
mathematica wrote:
how do you proove \lim_{x\to 0^+} e^{x\ln x} = e^{\lim_{x\to 0^+} x\ln x}


In short,

f continuous, \lim_{x\to x_0}g(x)=b

We want to show that \lim_{x\to x_0}f(g(x))=f(\lim_{x\to x_0}g(x))

Take any sequence {x_n} that x_n\to x_0. We have sequence {g(x_n)}, and \lim_{n\to \infty}g(x_n)=b

So \lim_{n\to \infty}f(g(x_n))=f(b)) because f is continuous at b.

PostPosted: Mon Sep 19, 2005 6:11 am  Back to top 
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mathematica
Poincare Conjecture
Poincare Conjecture


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#5
could you please use the epsilon-delta notation?

PostPosted: Tue Sep 20, 2005 9:18 pm  Back to top 
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blahblahblah
Birch & Swinnerton Dyer
Birch & Swinnerton Dyer

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#6
If you're just learning analysis (which seems likely), it would be a good exercise for you to prove some of the statements used here. I know of a proof that uses only the binomial theorem, but I don't have time to post it now (it's in Walter Rudin's Principles of Mathematical Analysis)

PostPosted: Tue Sep 20, 2005 11:17 pm  Back to top 
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