Graph all vertical and horizontal asymptotes of the rational function. \[ f(x)=\frac{7}{-x^{2}-8} \]

Function by following steps: - step0: Find the horizontal asymptotes: \(f\left(x\right)=\frac{7}{-x^{2}-8}\) - step1: Evaluate the limits \(\lim _{x\rightarrow +\infty}\left(f\left(x\right)\right)\) and \(\lim _{x\rightarrow -\infty}\left(f\left(x\right)\right):\) \(\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{7}{-x^{2}-8}\right)\\&\lim _{x\rightarrow -\infty}\left(\frac{7}{-x^{2}-8}\right)\end{align}\) - step2: Calculate: \(\begin{align}&0\\&0\end{align}\) - step3: The finite values are horizontal asymptotes: \(\begin{align}&f\left(x\right)=0\end{align}\) Find the vertical asymptotes of \( f(x)=\frac{7}{-x^{2}-8} \). Function by following steps: - step0: Find the vertical asymptotes: \(f\left(x\right)=\frac{7}{-x^{2}-8}\) - step1: The function has no vertical asymptote: \(\textrm{No vertical asymptotes}\) To analyze the rational function \[ f(x) = \frac{7}{-x^{2}-8} \] we can determine the asymptotes as follows: 1. **Horizontal Asymptotes:** - The horizontal asymptote is found by examining the behavior of \( f(x) \) as \( x \) approaches infinity or negative infinity. - In this case, as \( x \to \pm \infty \), the term \(-x^2\) dominates the denominator, leading to: \[ f(x) \to \frac{7}{-\infty} = 0 \] - Therefore, the horizontal asymptote is: \[ y = 0 \] 2. **Vertical Asymptotes:** - Vertical asymptotes occur where the denominator is zero and the numerator is not zero. - Setting the denominator equal to zero: \[ -x^2 - 8 = 0 \implies -x^2 = 8 \implies x^2 = -8 \] - Since \( x^2 = -8 \) has no real solutions, there are no vertical asymptotes. ### Summary of Asymptotes: - **Horizontal Asymptote:** \( y = 0 \) - **Vertical Asymptotes:** None You can graph the function and include the horizontal line \( y = 0 \) to represent the horizontal asymptote. There are no vertical lines to graph since there are no vertical asymptotes.