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

Feb 8, 2016

$x = \frac{\pi}{48} + \frac{\pi}{4} k$ with $k$ an integer.

Explanation:

$\tan$ has vertical asymptotes when the argument of the function is an odd multiple of $\frac{\pi}{2}$.

That is, we have vertical asymptotes when

$\frac{17 \pi}{12} + 4 x = \frac{\pi}{2} + \pi k$ $\text{ }$ ($k$ an integer)

$4 x = \frac{- 11 \pi}{12} + \pi k$

$4 x = \frac{- 11 \pi}{48} + \frac{\pi}{4} k$

Note that $\tan \left(\frac{- 11 \pi}{12}\right) = \tan \left(\frac{\pi}{12}\right)$ so there are vertical asymptotes at

$4 x = \frac{\pi}{12} + \pi k$ $\text{ }$ ($k$ an integer)

$x = \frac{\pi}{48} + \frac{\pi}{4} k$ with $k$ an integer.

(We could have left the solution as $x = \frac{- 11 \pi}{48} + \frac{\pi}{4} k$, but I like the positive reference number (angle).)