• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Infinite tetration and superroot of infinitesimal bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 02/26/2008, 09:26 AM Ivars Wrote:r= phi^(1/phi) , r= phi^(1/phi)^phi, r= phi^(1/phi)^phi^(1/phi) etc ... Now we think that infinity^(1/infinity) which was the beginning of this thread is not very well defined. ... So why not define the limits using the finite tangent angles? Or their trigonometric functions, e.g tangens? The one of the outgoing/incoming spiral for infinitesimal^(1/infinitesimal), the one of the limit circle at crossing for infinity^1/infinity. There are options to define those limits , why no one is used? 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: $\lim_{x\to\infty} x^{(1/x)^{x^{1/x}}}=1$ « 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

 Possibly Related Threads... Thread Author Replies Views Last Post [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 921 03/20/2018, 12:16 AM Last Post: tommy1729 [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 10,845 12/28/2014, 10:22 PM Last Post: MphLee Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 4,700 05/06/2014, 09:47 PM Last Post: tommy1729 Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 6,289 07/17/2013, 09:46 AM Last Post: Gottfried Wonderful new form of infinite series; easy solve tetration JmsNxn 1 4,085 09/06/2012, 02:01 AM Last Post: JmsNxn the infinite operator, is there any research into this? JmsNxn 2 5,115 07/15/2011, 02:23 AM Last Post: JmsNxn Infinite tetration of the imaginary unit GFR 40 50,122 06/26/2011, 08:06 AM Last Post: bo198214 Infinite Pentation (and x-srt-x) andydude 20 24,227 05/31/2011, 10:29 PM Last Post: bo198214 Infinite tetration fractal pictures bo198214 15 21,360 07/02/2010, 07:22 AM Last Post: bo198214 Infinite towers & solutions to Lambert W-function brangelito 1 3,512 06/16/2010, 02:50 PM Last Post: bo198214

Users browsing this thread: 1 Guest(s)