Questions: Find the horizontal and vertical asymptotes of (f(x)). [f(x)=frac4 x^2+11 x-3x^2+3 x-18] Find the horizontal asymptotes. Select the correct choice below and fill in any answer boxes within your choice. A. The horizontal asymptote(s) can be described by the line(s) (square) (Type an equation. Use a comma to separate answers as needed.) B. There are no horizontal asymptotes.

$Find the horizontal and vertical asymptotes of (f(x)). [f(x)=frac4 x^2+11 x-3x^2+3 x-18] Find the horizontal asymptotes. Select the correct choice below and fill in any answer boxes within your choice. A. The horizontal asymptote(s) can be described by the line(s) (square) (Type an equation. Use a comma to separate answers as needed.) B. There are no horizontal asymptotes.$

Transcript text: Find the horizontal and vertical asymptotes of $f(x)$. \[ f(x)=\frac{4 x^{2}+11 x-3}{x^{2}+3 x-18} \] Find the horizontal asymptotes. Select the correct choice below and fill in any answer boxes within your choice. A. The horizontal asymptote(s) can be described by the line(s) $\square$ (Type an equation. Use a comma to separate answers as needed.) B. There are no horizontal asymptotes.

Solution

Solution Steps

Step 1: Finding Horizontal Asymptotes

Since the degree of the numerator, $m = 2$, is equal to the degree of the denominator, $n = 2$, the horizontal asymptote is $y = 4$, where the leading coefficients of $P(x)$ and $Q(x)$ are 4 and 1 respectively.

Step 2: Finding Vertical Asymptotes

Solving $Q(x) = 0$ gives the potential vertical asymptotes. To ensure they are not holes, we check if these values also make the numerator zero. The values that do not make the numerator zero are the vertical asymptotes. The vertical asymptotes are at $x = -6, 3$.