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

By definition, arctanxarctan� is the inverse function of the restriction of the tangent function tantan to the interval (π2,π2)(-�2,�2) (see ).

The tangent function has vertical asymptotes x=π2�=-�2 and x=π2�=�2, for tanx=sinxcosxtan�=sin�cos� and cos±π2=0cos±�2=0.

Moreover, the graph of the inverse function f1�-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=±π2�=±�2 for y=tanx�=tan� correspond in this reflection to the horizontal asymptotes y=±π2�=±�2 for y=arctanx�=arctan�.

Here’s a graph of arctan(x):