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):
