Functions have asymptotes when they can be written as fractions of some sort and there are points in the domain where the denominator equals zero and the numerator does not equal zero. We need to rewrite f(x) = tan (x) as a fraction and look for these points.
We know from trigonometry that there is an identity that says
##f(x) = tan (x) = (sin(x))/(cos(x))##
We now look for the points where y = cos (x) = 0
The cosine function crosses the x axis (i.e., = 0) at odd multiples of ##pi/2##.
That is ##{. . . (-3*pi)/2, -pi/2, pi/2, (3*pi)/2, . . . }##
We verify that the function y = sin(x) does not cross the x-axis at those points, which it does not.
This means that at these points ##f(x)= tan(x)## has vertical asymptotes.
Note that this is exactly the same approach we use for finding vertical asymptotes on other functions that can be written as fractions. For example, ##g(x) = 1/x## has a vertical asymptote at the point x = 0 because the denominator equals zero while the numerator equals one.
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