Question

For what values of x, if any, does `f(x) = 1/((12x-9)sin(pi+(3pi)/x)` have vertical asymptotes?

Answer 1

`x=0,3/k,kinZZ` (where `k` is an integer)

Explanation:

Note that for a rational function such as the given function `f`, there will be a rational function whenever its denominator equals `0`.

So, there is a vertical asymptote whenever `(12x-9)sin(pi+(3pi)/x)=0`.

We can split this into two parts:

`{(12x-9=0),(sin(pi+(3pi)/x)=0):}`

The first can be solved to show that `x=9/12=3/4`.

The second is a little more difficult: note that `sin(x)=0` when `x=0,pi,2pi`, and so on. This can be written as `x=kpi`, where `kinZZ`, which means `k` is an integer.

Thus:

`pi+(3pi)/x=kpi" "" "," " "kinZZ`

Subtracting `pi`, we see that the right hand side remains `kpi`, because some integer multiple of `pi` minus `pi` is still an integer multiple of `pi`.

`(3pi)/x=kpi" "" "," " " "kinZZ`

Rearranging:

`3pi=x(kpi)" "" "," "" "kinZZ`

`x=(3pi)/(kpi)=3/k" "" "," "" "kinZZ`

Note that the previous solution from `12x-9=0` gave `x=3/4`, which is included in the solution `x=3/k,kinZZ`.

Furthermore, since `(3pi)/x` is included within the sine function, there will be a vertical asymptote at `x=0`.