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 Infinite tetration and superroot of infinitesimal Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 02/26/2008, 09:31 PM bo198214 Wrote:I dont understand why you invoke a polar graph here, the limits are very well defined and well-known: $\lim_{x\to\infty} x^{1/x} = 1$ $\lim_{x\to 0} x^{1/x} = 0$ You can already guess it when you look at the (cartesian) graph of the selfroot. $\lim_{x\to\infty} x^{(1/x)^x} = \lim_{x\to\infty} x^{1/x^x} = \lim_{x\to\infty} e^{\ln(x)/x^x}=e^{\lim_{x\to\infty}\ln(x)/x^x }=e^0=1$ $\lim_{x\to\infty} x^{(1/x)^{x^{1/x}}}=\exp\left(\ln(x)\frac{1 }{x^{x^{1/x}}}\right)$ Here we know that $x^{1/x}>1$ for $x>1$ and hence $x^{x^{1/x}}>x$ for $x>1$ and hence $\lim_{x\to\infty}\ln(x)/x^{x^{1/x}}=0$. Finally again: $\lim_{x\to\infty} x^{(1/x)^{x^{1/x}}}=1$ Hi, I reread the beginning of the thread and it was dx^(1/dx) that was a problem, so lim x^1/x when x-> 0 is undefined in real numbers - that I understood from both jaydfox and Andy. Since in my understanding infinity is 1/infinitesimal, x-> infinity is perhaps complementary to first one. When You add another x making x^(1/x)^x You perhaps forcibly drive the expression into the limit You wish as exponent is higher level of infinitesimal than x is infinity. It does not prove (1/dx)^dx=1- these infinitesimals must be of same order. I chose polar coordinates because there You can see what infinity of self root looks like - it is a unit circle-all 4 expressions tend to it, though from different sides. See attached file. As to self root close to 0, dx^1/dx, 2 of selfroots seem to start at 0, 2-at infinity, depending on combination of signs, so this is still undecided for me. Attached Files Image(s) « Next Oldest | Next Newest »

 Messages In This Thread Infinite tetration and superroot of infinitesimal - by Ivars - 12/16/2007, 06:02 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 12/17/2007, 02:27 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/18/2007, 02:59 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/19/2007, 08:42 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/19/2007, 04:03 PM RE: Infinite tetration and superroot of infinitesimal - by jaydfox - 12/19/2007, 06:29 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/19/2007, 08:55 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/20/2007, 08:23 AM RE: Infinite tetration and superroot of infinitesimal - by andydude - 12/20/2007, 06:31 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 12/20/2007, 03:16 AM RE: Infinite tetration and superroot of infinitesimal - by jaydfox - 12/20/2007, 07:25 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/20/2007, 09:02 PM RE: Infinite tetration and superroot of infinitesimal - by jaydfox - 12/20/2007, 09:08 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/20/2007, 09:50 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 12/21/2007, 02:08 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/21/2007, 05:49 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 12/21/2007, 08:06 PM RE: Infinite tetration and superroot of infinitesimal - by jaydfox - 12/21/2007, 08:40 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/21/2007, 10:41 PM RE: Infinite tetration and superroot of infinitesimal - by jaydfox - 12/21/2007, 11:00 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/24/2007, 09:56 AM RE: Infinite tetration and superroot of infinitesimal - by jaydfox - 12/24/2007, 04:40 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/25/2007, 09:31 AM RE: Infinite tetration and superroot of infinitesimal - by andydude - 12/25/2007, 06:46 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/25/2007, 09:37 PM RE: Infinite tetration and superroot of infinitesimal - by jaydfox - 12/26/2007, 09:27 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/04/2008, 06:51 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/04/2008, 07:08 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/04/2008, 09:48 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/05/2008, 07:37 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/05/2008, 07:49 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/05/2008, 10:30 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/06/2008, 01:10 AM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/06/2008, 09:42 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/06/2008, 04:54 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/06/2008, 09:26 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 12/28/2007, 12:31 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/02/2008, 08:56 AM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/02/2008, 10:34 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/02/2008, 11:47 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/05/2008, 09:38 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/07/2008, 10:04 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/08/2008, 08:28 AM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/08/2008, 08:52 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/08/2008, 03:44 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/09/2008, 12:50 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/09/2008, 01:53 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/08/2008, 03:41 PM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/08/2008, 11:44 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/10/2008, 09:19 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/13/2008, 01:14 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/15/2008, 01:09 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/23/2008, 11:25 AM RE: Infinite tetration and superroot of infinitesimal - by andydude - 01/25/2008, 10:52 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/23/2008, 05:13 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/23/2008, 07:52 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/23/2008, 09:22 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/24/2008, 03:44 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/25/2008, 03:13 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/25/2008, 11:36 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/26/2008, 02:13 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/26/2008, 04:01 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/27/2008, 01:03 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/27/2008, 12:24 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/27/2008, 06:56 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/27/2008, 08:51 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/28/2008, 02:25 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/28/2008, 03:36 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/28/2008, 09:49 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/28/2008, 10:29 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/30/2008, 09:56 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/30/2008, 04:09 PM RE: Infinite tetration and superroot of infinitesimal - by bo198214 - 01/30/2008, 05:09 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/30/2008, 07:28 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/30/2008, 08:21 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/31/2008, 07:41 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/31/2008, 11:13 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 01/31/2008, 03:56 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 01/31/2008, 05:06 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/01/2008, 07:42 AM RE: Infinite tetration and superroot of infinitesimal - by GFR - 02/01/2008, 10:31 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/01/2008, 11:14 AM RE: Infinite tetration and superroot of infinitesimal - by GFR - 02/01/2008, 08:56 PM RE: Infinite tetration and superroot of infinitesimal - by bo198214 - 02/01/2008, 09:29 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/01/2008, 10:00 PM RE: Infinite tetration and superroot of infinitesimal - by GFR - 02/02/2008, 03:31 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/05/2008, 10:58 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/06/2008, 07:43 PM RE: Infinite tetration and superroot of infinitesimal - by bo198214 - 02/26/2008, 09:26 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/26/2008, 09:31 PM RE: Infinite tetration and superroot of infinitesimal - by bo198214 - 02/26/2008, 10:23 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/27/2008, 09:36 AM RE: Infinite tetration and superroot of infinitesimal - by bo198214 - 02/27/2008, 03:14 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 02/27/2008, 09:00 PM RE: Infinite tetration and superroot of infinitesimal - by bo198214 - 03/13/2008, 08:30 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 04/05/2008, 06:42 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 05/10/2008, 08:33 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 06/27/2008, 04:51 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/04/2008, 07:02 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/04/2008, 09:01 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/05/2008, 08:12 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 10/16/2008, 10:39 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/05/2008, 11:26 AM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/10/2008, 12:26 PM RE: Infinite tetration and superroot of infinitesimal - by Ivars - 07/11/2008, 09:27 AM

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