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

To determine the horizontal and vertical asymptotes of the function \( f(x) = \frac{x^{8}}{x^{4} + 2} \), let's analyze each type of asymptote separately. ### Horizontal Asymptotes **Method:** 1. **Determine the degrees of the numerator and the denominator:** - **Numerator:** \( x^8 \) has a degree of 8. - **Denominator:** \( x^4 + 2 \) has a degree of 4. 2. **Compare the degrees:** - 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 \( \pm\infty \), \( f(x) \) increases without bound. **Conclusion:** - **Choice C:** The function has **no horizontal asymptotes**. ### Vertical Asymptotes **Method:** 1. **Find the values of \( x \) that make the denominator zero:** \[ x^4 + 2 = 0 \implies x^4 = -2 \] 2. **Solve for \( x \):** - The equation \( x^4 = -2 \) has **no real solutions** because \( x^4 \) is always non-negative for real numbers, and it cannot equal a negative number. **Conclusion:** - **Choice C:** The function has **no vertical asymptotes**. ### Summary of Asymptotes: - **Horizontal Asymptotes:** **Choice C.** The function has **no horizontal asymptotes**. - **Vertical Asymptotes:** **Choice C.** The function has **no vertical asymptotes**. ### Final Answer: **Horizontal Asymptotes:** C. The function has no horizontal asymptotes. **Vertical Asymptotes:** C. The function has no vertical asymptotes.