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

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: \(\textrm{No vertical asymptotes}\) Find the horizontal asymptotes of \( f(x)=\frac{7}{-x^{2}-8} \). 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: \(\begin{align}&f\left(x\right)=0\end{align}\) 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 the function as \(x\) approaches infinity or negative infinity. For rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). - In this case, the degree of the numerator (which is 0) is less than the degree of the denominator (which is 2). Therefore, the horizontal asymptote is: \[ y = 0 \] 2. **Vertical Asymptotes**: - Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points). We set the denominator equal to zero: \[ -x^{2} - 8 = 0 \] This simplifies to: \[ -x^{2} = 8 \quad \Rightarrow \quad x^{2} = -8 \] Since \(x^{2} = -8\) has no real solutions (as the square of a real number cannot be negative), there are no vertical asymptotes. ### Summary of Asymptotes: - **Horizontal Asymptote**: \(y = 0\) - **Vertical Asymptotes**: None Thus, the function has a horizontal asymptote at \(y = 0\) and no vertical asymptotes.