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

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 finding inverse functions ), 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):