How do you find a vertical asymptote for y = tan(x)?

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Functions feel asymptotes when they can be written as refractory of some kind and there are purposes in the lordship where the denominator similars nothing and the numerator does not similar nothing. We deficiency to rewrite f(x) = tan (x) as a faction and behold for these purposes.

We distinguish from trigonometry that there is an personality that says

##f(x) = tan (x) = (sin(x))/(cos(x))##

We now behold for the purposes where y = cos (x) = 0

The cosine business ill-conditionedes 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 warrant that the business y = sin(x) does not ill-conditioned the x-axis at those purposes, which it does not.

This resources that at these purposes ##f(x)= tan(x)## has perpendicular asymptotes.

Note that this is precisely the similar advent we use for decision perpendicular asymptotes on other businesss that can be written as refractory. For copy, ##g(x) = 1/x## has a perpendicular asymptote at the purpose x = 0 consequently the denominator similars nothing conjuncture the numerator similars one.

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