Question

For what values of x, if any, does `f(x) = tan((17pi)/12+4x)` have vertical asymptotes?

Answer 1

`x=pi/48+pi/4k` with `k` an integer.

Explanation:

`tan` has vertical asymptotes when the argument of the function is an odd multiple of `pi/2`.

That is, we have vertical asymptotes when

`(17pi)/12+4x = pi/2 + pik` `" "` (`k` an integer)

`4x = (-11pi)/12+pik`

`4x=(-11pi)/48 + pi/4k`

Note that `tan((-11pi)/12) = tan(pi/12)` so there are vertical asymptotes at

`4x = pi/12 + pi k` `" "` (`k` an integer)

`x=pi/48+pi/4k` with `k` an integer.

(We could have left the solution as `x=(-11pi)/48 + pi/4k`, but I like the positive reference number (angle).)