Question

question
find all vertical asymptotes of the following function.
$f(x) = \frac{x^2 + 8x}{3x^2 - 75}$
answer attempt 1 out of 2
no vertical asymptotes
no vertical asymptotes
one vertical asymptote
two vertical asymptotes

Explanation:

Step1: Factor numerator and denominator

Factor numerator: \(x^2 + 8x = x(x + 8)\)
Factor denominator: \(3x^2 - 75 = 3(x^2 - 25) = 3(x - 5)(x + 5)\)
So \(f(x)=\frac{x(x + 8)}{3(x - 5)(x + 5)}\)

Step2: Find values that make denominator zero

Set denominator \(3(x - 5)(x + 5)=0\)
Solve: \(x - 5 = 0\) or \(x + 5 = 0\)
\(x = 5\) or \(x = -5\)

Step3: Check if numerator is zero at these x-values

For \(x = 5\): Numerator \(5(5 + 8)=65
eq0\)
For \(x = -5\): Numerator \(-5(-5 + 8)=-15
eq0\)

Since denominator is zero and numerator is non - zero at \(x = 5\) and \(x=-5\), there are two vertical asymptotes.

Answer:

Two Vertical Asymptotes