Question

For what values of x, if any, does `f(x) = 1/((x-9)(x-1)(x-5))` have vertical asymptotes?

Answer 1

`x=9,x=3,x=5`

Explanation:

To find the vertical asymptotes, you simply have to find the values of `x` where the bottom of the fraction will be equal `0`.

In this case, we can use the equation: `(x-9)(x-1)(x-5)=0` for which there are 3 possible solutions.

These are: `x=9,x=3,x=5`.

So we will have vertical asymptotes here. To help demonstrate this we can graph the equation:

graph{1/((x-3)(x-5)(x-9) [-3.73, 16.27, -4.48, 5.52]}

As you can see the graph shoots off to infinity and does not cross at the vertical lines `x=3,x=5,x=9`

As a point of interest notice there is one horizontal asymptote at `x=0`.