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As
is odd, we have that the line whose equation is
is a an asymptote to
; by periodicity we conclude that all the lines whose equations are
, for integer
, are asymptotes to
.
Part of the graph of this function is displayed in Figure 11.
At this point, we wish to mention a very common error: it is not sufficient for a function not to be defined at a single point
for the line whose equation is
to be an asymptote to the graph of
. For example, let
be given by
. Then
, and the
axis is not an asymptote to the graph of
. See Figure 12.
We said already that an infinite one-sided limit at
shows the existence of a vertical asymptote whose equation is
. In example function discussion rational function of exp we will see a situation where the function has two different one-sided limits at one point: the one is finite, the other is infinite, showing the existence of a vertical asymptote.
Noah Dana-Picard 2007-12-28