Question

question
find all vertical asymptotes of the following function.
$f(x) = \frac{5x^2 - 30x}{3x - 18}$
answer attempt 1 out of 2
no vertical asymptotes

Explanation:

Step1: Simplify the function

First, factor the numerator and the denominator. The numerator \(5x^2 - 30x\) can be factored as \(5x(x - 6)\), and the denominator \(3x - 18\) can be factored as \(3(x - 6)\). So the function becomes:

$$f(x)=\frac{5x(x - 6)}{3(x - 6)}$$

Then, we can cancel out the common factor \((x - 6)\) (note that \(x
eq6\) to avoid division by zero in the original function), so the simplified function is \(f(x)=\frac{5x}{3}\) (a linear function), except at \(x = 6\) where the original function is undefined. But since the discontinuity at \(x = 6\) is a removable discontinuity (a hole) and not a vertical asymptote (vertical asymptotes occur where the denominator is zero and the numerator is non - zero at that point), there are no vertical asymptotes.

Answer:

No Vertical Asymptotes