Question

question
find all vertical asymptotes of the following function.
f ( x ) = \frac { 3 x ^ { 2 } - 108 } { x - 9 }
answer attempt 1 out of 2
no vertical asymptotes
no vertical asymptotes
one vertical asymptote
two vertical asymptotes

Explanation:

Step1: Simplify the function

Factor the numerator \(3x^{2}-108 = 3(x^{2}-36)=3(x + 6)(x - 6)\). So the function becomes \(f(x)=\frac{3(x + 6)(x - 6)}{x - 9}\).

Step2: Analyze the denominator

The denominator \(x-9 = 0\) when \(x = 9\).

Step3: Check the limit

\(\lim_{x
ightarrow9}\frac{3x^{2}-108}{x - 9}=\lim_{x
ightarrow9}\frac{3(x^{2}-36)}{x - 9}=\lim_{x
ightarrow9}\frac{3(x + 6)(x - 6)}{x - 9}\). Substituting \(x = 9\) gives \(\frac{3(9 + 6)(9 - 6)}{9 - 9}=\frac{3\times15\times3}{0}\), and the limit is either \(\infty\) or \(-\infty\) (the function approaches infinity or negative infinity as \(x\) approaches \(9\)).

Answer:

One Vertical Asymptote (\(x = 9\))