By definition,is the inverse function of the restriction of the tangent function to the interval (see ).
The tangent function has vertical asymptotesand , for and .
Moreover, the graph of the inverse functionof a one-to-one function is obtained from the graph of by reflection about the line (see ), which transforms vertical lines into horizontal lines.
Thus, the vertical asymptotesfor correspond in this reflection to the horizontal asymptotes for .
Here’s a graph of arctan(x):