Find the horizontal and vertical asymptotes of \( f(x) \). \( I(x)=\frac{x^{8}}{x^{4}+2} \) Find the horizontal asymptotos. Seloct the correct choice and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is \( \square \) and the bottom asymptote is (Type equations.) C. The function has no horizontal asymptotes.

To determine the horizontal asymptotes of the function \( I(x) = \frac{x^{8}}{x^{4}+2} \), we'll analyze the behavior of \( I(x) \) as \( x \) approaches both \( +\infty \) and \( -\infty \). ### Steps to Determine Horizontal Asymptotes: 1. **Compare Degrees of Numerator and Denominator:** - **Numerator Degree:** 8 (from \( x^8 \)) - **Denominator Degree:** 4 (from \( x^4 \)) 2. **Determine the Relationship:** - Since the degree of the numerator (8) is **greater** than the degree of the denominator (4), the function does **not** have a horizontal asymptote. Instead, as \( x \) approaches \( \pm\infty \), \( I(x) \) will grow without bound. 3. **Conclusion:** - **No Horizontal Asymptote Exists** for this function because the function increases indefinitely as \( x \) approaches both \( +\infty \) and \( -\infty \). ### Correct Choice: **C. The function has no horizontal asymptotes.**