Wikipedia:Reference desk/Archives/Mathematics/2018 October 10

= October 10 =

Limit question
Is there a nice way to show that $$\lim_{x \to \infty} (\csc \frac{\pi}{x + 1} - \csc \frac{\pi}{x}) = \frac{1}{\pi}$$? Trying to turn it into 0/0 and using L'Hôpital's rule seems to just get me stuck in a mess of algebra that doesn't seem to be getting me any closer to the nice answer. Double sharp (talk) 14:57, 10 October 2018 (UTC)


 * One way is to first make the substitution $$u = 1/x$$ and then let $$u \to 0^+.$$ Then, you can write each cosecant as its Laurent series around 0: $$\csc u = \frac{1}{u} + \frac{u}{6} + O(u^3).$$  Then you can take the limit easily.  –Deacon Vorbis (carbon &bull; videos) 15:13, 10 October 2018 (UTC)
 * (Or better, all that matters here is $$\csc u = 1/u + O(u)$$). –Deacon Vorbis (carbon &bull; videos) 15:21, 10 October 2018 (UTC)
 * Thank you for your help; that indeed worked perfectly! Clearly I have some more reading to do. ^_^ Double sharp (talk) 15:24, 10 October 2018 (UTC)
 * I hope you don't mind that I linked to your answer elsewhere, as this limit came up in a forum discussion I was involved in today. ^_^ Double sharp (talk) 15:44, 10 October 2018 (UTC)
 * Naw, no worries. –Deacon Vorbis (carbon &bull; videos) 15:59, 10 October 2018 (UTC)
 * Thank you! Double sharp (talk) 16:02, 10 October 2018 (UTC)