a curious limit  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) + Thread: a curious limit (/showthread.php?tid=623) Pages:
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a curious limit  JmsNxn  04/14/2011 I'm wondering if the following limit is nonzero; v E R and if so, what is it equal to? Thanks I know it doesn't converge for RE: a curious limit  nuninho1980  04/14/2011 attention: h > o is bad but yes h > 0. because 'o' isn't number and is letter. lol RE: a curious limit  JmsNxn  04/14/2011 (04/14/2011, 08:10 PM)nuninho1980 Wrote: attention: h > o is bad but yes h > 0. because 'o' isn't number and is letter. lol no no, I put 0, but latex just designs it to look like o RE: a curious limit  bo198214  04/14/2011 (04/14/2011, 08:01 PM)JmsNxn Wrote: I'm wondering if the following limit is nonzero; v E R The powers with noninteger exponents are not uniquely defined in the complex plane. In your case you would need to put: But then the standard logarithm has a cut on , which is quite arbitrary: one could put a cut however one likes. For example could spiral around 0, while moving towards 0 and would increase/decrease its imaginary part by in each round. I guess it really depends on how approaches 0. RE: a curious limit  JmsNxn  04/14/2011 let's take the limit from positive (keep it simple first) so: Is there any way of reexpressing this limit? RE: a curious limit  bo198214  04/15/2011 (04/14/2011, 10:55 PM)JmsNxn Wrote: let's take the limit from positive (keep it simple first) But then its not difficult, since on the reals, the whole limit goes to (complex) except for , for which the whole limit is 0. RE: a curious limit  JmsNxn  04/15/2011 (04/15/2011, 07:14 AM)bo198214 Wrote: But then its not difficult, since on the reals, the whole limit goes to (complex) except for , for which the whole limit is 0. Alright, how about where is taken to mean approaching along the axis. I think it's the equivalent of: which I guess converges to negative infinity again, except for 1e^{vi}=0 hmm, seems this is less interesting than I thought. RE: a curious limit  JmsNxn  04/16/2011 is there any way of letting h approach zero such that: ? RE: a curious limit  bo198214  04/16/2011 (04/16/2011, 07:22 PM)JmsNxn Wrote: is there any way of letting h approach zero such that: The logarithm of is . So regardless how you approach 0, i.e. , you will allways have that . So the answer is no (except ). RE: a curious limit  JmsNxn  04/16/2011 (04/16/2011, 07:41 PM)bo198214 Wrote: The logarithm of is . that's what I thought 