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

##### 1 Answer

#### Explanation:

Note that for a rational function such as the given function

So, there is a vertical asymptote whenever

We can split this into two parts:

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

The first can be solved to show that

The second is a little more difficult: note that

Thus:

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

Subtracting

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

Rearranging:

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

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

Note that the previous solution from

Furthermore, since *within* the sine function, there will be a vertical asymptote at