question\nfind all vertical asymptotes of the following function.\n f ( x ) = \frac { 3 x ^ { 2 } - 108 } {…

question\nfind all vertical asymptotes of the following function.\n f ( x ) = \frac { 3 x ^ { 2 } - 108 } { x - 9 } \nanswer attempt 1 out of 2\nno vertical asymptotes\nno vertical asymptotes\none vertical asymptote\ntwo vertical asymptotes

question\nfind all vertical asymptotes of the following function.\n f ( x ) = \frac { 3 x ^ { 2 } - 108 } { x - 9 } \nanswer attempt 1 out of 2\nno vertical asymptotes\nno vertical asymptotes\none vertical asymptote\ntwo vertical asymptotes

Answer

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\rightarrow9}\frac{3x^{2}-108}{x - 9}=\lim_{x\rightarrow9}\frac{3(x^{2}-36)}{x - 9}=\lim_{x\rightarrow9}\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))