While working on some probcapability question, I had to evaluate $lim_x o infty arctan(x)$. I knew the answer intuitively as $pi/2$, yet I cannot figure out just how to prove it by elementary indicates (without resorting to $epsilon-delta$ arguments). How does one prove it (preferably, without resorting to L"Hopital"s rule)?




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The $arctan$ attribute is the inverse feature of $$ an:left(-fracpi2,fracpi2 ight) ightarrowBbb R$$and also considering that this attribute is monotonically raising then$$lim_x ofracpi 2 an x=+inftyiff lim_x o+inftyarctan x=fracpi2$$


$$lim_x o +infty arctan x = fracπ2$$ and $$lim_x o -infty arctan x = -fracπ2,$$then in total$$lim_x o infty arctan x = fracπ2 slrfc.orgrmsgn(x).$$


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