How do you find horizontal asymptotes for ##f(x) = arctan(x)## ?

By definition, ##arctan x## is the inverse function of the restriction of the tangent function ##tan## to the interval ##(-pi/2,pi/2)## (see ).

The tangent function has vertical asymptotes ##x=-pi/2## and ##x=pi/2##, for ##tan x=sin x/cos x## and ##cos \pm pi/2=0##.

Moreover, the graph of the inverse function ##f^(-1)## of a one-to-one function ##f## is obtained from the graph of ##f## by reflection about the line ##y=x## (see ), which transforms vertical lines into horizontal lines.

Thus, the vertical asymptotes ##x=\pm pi/2## for ##y=tan x## correspond in this reflection to the horizontal asymptotes ##y=\pm pi/2## for ##y=arctan x##.

Here's a graph of arctan(x):