# For what values of x, if any, does f(x) = 1/((12x-8)sin(pi+(3pi)/x)  have vertical asymptotes?

May 2, 2018

The function has vertical asymptotes in the following points:

• $x = \frac{2}{3}$
• All the points of the form $\setminus \frac{3}{2 k - 1}$
• All the points of the form $\setminus \frac{3}{2 k}$

Note that, since $k$ can be any integer number, the last two points can be summed up in

• All the points of the form $\setminus \frac{3}{n}$, with $n \setminus \in \setminus m a t h \boldsymbol{Z}$

#### Explanation:

A function usually has vertical asymptotes where it is not defined, either because of a zero denominator, or a zero argument in a logarithm.

In this case, we have to look for zeros of the denominator.

The denominator, in turn, is a multiplication of two factors, namely $\left(12 x - 8\right)$ and $\sin \left(\setminus \pi + \setminus \frac{3 \setminus \pi}{x}\right)$. A multiplication of two factor is zero if either of the two factors is zero, so let's examine them one by one.

$12 x - 8$ equals zero if and only if...

$12 x - 8 = 0 \setminus \iff 12 x = 8 \setminus \iff x = \frac{8}{12} = \frac{2}{3}$

$\sin \left(\setminus \pi + \setminus \frac{3 \setminus \pi}{x}\right)$ equals zero if and only if the argument equals $2 k \setminus \pi$ or $\setminus \pi + 2 k \setminus \pi$. Let's split the two cases again:

First case: the argument equals $2 k \setminus \pi$

$\setminus \pi + \setminus \frac{3 \setminus \pi}{x} = 2 k \setminus \pi \setminus \iff \setminus \frac{3 \setminus \pi}{x} = 2 k \setminus \pi - \setminus \pi \setminus \iff \setminus \frac{x}{3 \setminus \pi} = \setminus \frac{1}{2 k \setminus \pi - \setminus \pi}$

$\setminus \iff x = \setminus \frac{3 \setminus \pi}{2 k \setminus \pi - \setminus \pi} = \setminus \frac{3}{2 k - 1}$

Second case: the argument equals $\setminus \pi + 2 k \setminus \pi$

$\setminus \pi + \setminus \frac{3 \setminus \pi}{x} = \setminus \pi + 2 k \setminus \pi \setminus \iff \setminus \frac{3 \setminus \pi}{x} = 2 k \setminus \pi \setminus \iff \setminus \frac{x}{3 \setminus \pi} = \setminus \frac{1}{2 k \setminus \pi}$

$\setminus \iff x = \setminus \frac{3 \setminus \pi}{2 k \setminus \pi} = \setminus \frac{3}{2 k}$