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I think that by putting some of your above statements together, we've got all the bits we need to prove that 2.999999... = 3. The key one was where you said that you were writing 0.0000....000001 as 1/(10^n) (or 10^(-n)) with n approaching infinity. And as Faldage correctly stated, that's equal to zero. So now put it all together:
2.99999999... = lim (n->inf) (3 - 10^(-n)) - this is more or less what was stated above, right? That it differs from 3 by a small amount, namely 10^(-n).
But using the rules of limits, which have been implicitly used above, so we must be in agreement on them, re-write the RHS, so now we have
2.99999999... = lim (n->inf) (3) - lim (n->inf) (10^(-n))
Again using the rules of limits, the limit of a constant (the first term) is a constant itself, and the limit of the second term is, as Faldage said, zero, so we have
2.99999999... = 3 - 0
2.99999999... = 3
Making up an entirely different function from the one being discussed that shows that the limit OF THE PARTICULAR FUNCTION is different when approaching from above or below has no relevance to this argument, which is "how do we write the number 3?", not "can we think of a function that has a limit that is undefined at 3?". Certainly f(x) = 3/(x-3) is undefined at x = 3 but that doesn't tell you a lot about the point x = 3 on the number line.
Entire Thread Subject Posted By Posted dr rav 01/21/03 05:27 PM Re: dr wentworth 01/21/03 05:36 PM Re: dr Faldage 01/21/03 05:42 PM Re: dr Wordwind 01/21/03 06:04 PM Re: dr wwh 01/21/03 06:17 PM Re: Dr or Dr. wwh 01/21/03 06:20 PM Re: dr Faldage 01/21/03 06:33 PM Re: dr wwh 01/21/03 06:54 PM Re: dr Faldage 01/21/03 07:23 PM Re: dr wwh 01/21/03 07:43 PM Re: dr AnnaStrophic 01/21/03 11:25 PM Re: dr rav 01/22/03 01:55 PM Re: dr TheFallibleFiend 01/21/03 09:43 PM Re: dr johnjohn 01/21/03 11:25 PM Re: dr wwh 01/22/03 01:34 AM Re: dr Faldage 01/22/03 11:05 AM The RhubarbCommando speaks out for Britain. RhubarbCommando 01/22/03 12:24 PM Re: dr boronia 01/22/03 02:03 PM Post deleted by Wordwind Wordwind 01/22/03 02:10 PM Re: dr rkay 01/23/03 09:19 AM Re: dr Capfka 01/23/03 11:19 AM Re: dr Bean 01/22/03 02:55 PM Re: dr TheFallibleFiend 01/22/03 03:14 PM Re: dr Bean 01/22/03 04:05 PM Re: dr TheFallibleFiend 01/22/03 07:02 PM Re: dr Faldage 01/22/03 07:32 PM Re: dr TheFallibleFiend 01/22/03 08:23 PM Re: dr Faldage 01/22/03 08:29 PM Re: dr Bean 01/23/03 11:34 AM math emanuela 01/28/03 04:54 PM answer emanuela 01/28/03 04:55 PM Re: answer Faldage 01/28/03 06:38 PM Re: answer wwh 01/28/03 06:46 PM Re: answer Faldage 01/28/03 07:08 PM Re: answer wwh 01/28/03 07:45 PM Re: answer Faldage 01/28/03 07:53 PM Re: answer Alex Williams 01/28/03 10:38 PM Re: answer Bean 01/28/03 11:46 PM Re: answer tsuwm 01/29/03 12:11 AM Post deleted by Wordwind Wordwind 01/29/03 12:16 AM e Alex Williams 01/29/03 02:04 AM Re: e Faldage 01/29/03 11:05 AM Re: e Bean 01/29/03 11:10 AM Re: e Bean 01/29/03 11:14 AM Re: e TheFallibleFiend 01/29/03 11:50 AM Re: e Alex Williams 01/29/03 01:48 PM Re: e TheFallibleFiend 01/29/03 02:10 PM Re: e Faldage 01/29/03 02:36 PM Re: e Bean 01/29/03 02:49 PM Re: one = non-one tsuwm 01/29/03 02:58 PM Re: one = non-one Bean 01/29/03 03:12 PM Re: e Faldage 01/29/03 03:16 PM Re: e Bean 01/29/03 03:54 PM Re: e Alex Williams 01/29/03 04:13 PM Re: e tsuwm 01/29/03 04:18 PM Re: e Faldage 01/29/03 04:33 PM Re: e Alex Williams 01/29/03 05:04 PM Re: e Buffalo Shrdlu 01/29/03 05:32 PM Re: e Alex Williams 01/29/03 06:04 PM Re: e Faldage 01/29/03 06:06 PM infinity emanuela 01/30/03 06:36 AM Re: infinity Alex Williams 01/30/03 10:40 AM Re: infinity Bean 01/30/03 11:24 AM Re: numbers real and rational Faldage 01/30/03 12:03 PM Re: numbers real and rational Alex Williams 01/30/03 01:01 PM Re: numbers real and rational emanuela 01/30/03 01:56 PM Re: numbers real and rational Faldage 01/30/03 02:00 PM . Alex Williams 01/30/03 03:22 PM Re: . Faldage 01/30/03 03:31 PM Re: e Bean 01/29/03 04:32 PM Re: e Bean 01/29/03 02:45 PM math is not an exact science wofahulicodoc 01/30/03 03:31 PM numbers emanuela 01/30/03 04:54 PM Re: math Faldage 01/28/03 05:15 PM Re: math wwh 01/28/03 06:00 PM Re: math musick 01/28/03 06:16 PM parallel lines emanuela 01/29/03 01:31 PM 0,99999..... emanuela 01/29/03 01:17 PM Re: dr vika 01/28/03 04:42 PM Re: dr rav 01/22/03 08:51 PM Re: dr modestgoddess 01/23/03 08:32 PM Re: dr Faldage 01/23/03 08:40 PM Re: dr modestgoddess 01/23/03 08:44 PM Re: dr Bean 01/24/03 12:09 AM Re: dr Capfka 01/24/03 08:46 PM Re: dr Faldage 01/27/03 07:56 PM Re: dr AnnaStrophic 01/27/03 07:58 PM Re: duhing … and three dots Faldage 01/27/03 08:08 PM Re: duhing … and three dots wwh 01/27/03 08:15 PM Re: duhing … and three dots Faldage 01/27/03 08:18 PM Re: dr tsuwm 01/27/03 08:04 PM
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