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

##### 1 Answer

The function has vertical asymptotes in the following points:

#x = 2/3# - All the points of the form
#\frac{3}{2k-1}# - All the points of the form
#\frac{3}{2k}#

Note that, since

- All the points of the form
#\frac{3}{n}# , with#n \in \mathbb{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

**First case: the argument equals #2k\pi#**

**Second case: the argument equals #\pi+2k\pi#**