Using the asymptotes concept, it is found that:
- The vertical asymptote is at x = 9.
- The horizontal asymptote is at y = 3. Hence, the end behavior of the function is y = 3.
What are the asymptotes of a function f(x)?
- The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator.
- The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.
In this problem, the function is:
[tex]f(x) = \frac{3x}{x - 9}[/tex]
As for the vertical asymptote, we have that:
[tex]x - 9 \neq 0 \rightarrow x \neq 9[/tex]
Hence, there is a vertical asymptote is at x = 9.
For the horizontal, we have that:
[tex]\lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{3x}{x - 9} = \lim_{x \rightarrow \infty} \frac{3x}{x} = \lim_{x \rightarrow \infty} 3 = 3[/tex]
The horizontal asymptote is at y = 3. Hence, the end behavior of the function is y = 3.
More can be learned about asymptotes at https://brainly.com/question/16948935
