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

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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)

@Double sharp: 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 • videos) 15:13, 10 October 2018 (UTC)

(Or better, all that matters here is $\csc u=1/u+O(u)$). –Deacon Vorbis (carbon • videos) 15:21, 10 October 2018 (UTC)

@Deacon Vorbis: 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)

@Deacon Vorbis: 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 • videos) 15:59, 10 October 2018 (UTC)

Thank you! Double sharp (talk) 16:02, 10 October 2018 (UTC)