Question

For what values of x, if any, does `f(x) = 1/((x-2)(x-2)(e^x-3))` have vertical asymptotes?

Answer 1

`x=2` and `x=ln3`

Explanation:

Vertical asymptotes occur when the denominator of a function equals `0`.

`(x-2)(x-2)(e^x-3)=0`

`(x-2)^2(e^x-3)=0`

Like when factoring and solving a quadratic equation, when we have multiple terms being multiplied by one another and equalling `0`, any of the terms can be equal to `0` for the expression. Thus, we can set both of the multiplied equal to `0`.

`mathbf((1))`

`(x-2)^2=0`

`x-2=0`

`x=2`

`mathbf((2))`

`e^x-3=0`

`e^x=3`

`x=ln3`

The vertical asymptotes occur at `x=2` and `x=ln3`.