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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)[reply]

@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)[reply]

(Or better, all that matters here is

\csc u=1/u+O(u)

). –Deacon Vorbis (carbon • videos) 15:21, 10 October 2018 (UTC)[reply]

@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)[reply]

@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)[reply]

Naw, no worries. –Deacon Vorbis (carbon • videos) 15:59, 10 October 2018 (UTC)[reply]

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