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.
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Bonus Knowledge
To find the horizontal asymptotes of the function \( f(x) = \frac{x^8}{x^4 + 2} \), we assess the degrees of the numerator and denominator. The degree of the numerator is 8, while the degree of the denominator is 4. Since the degree of the numerator is greater than the degree of the denominator, there are no horizontal asymptotes. So, the correct choice is C: The function has no horizontal asymptotes. For vertical asymptotes, we need to determine where the function goes to infinity, which occurs when the denominator is zero. Setting \( x^4 + 2 = 0 \) gives us \( x^4 = -2 \); however, this has no real solutions as the left side is always non-negative. Therefore, the function has no vertical asymptotes. The correct choice is C: The function has no vertical asymptotes.
