Find the vertical, horizontal, and oblique asymptotes, if any, of the given rational function. \[ H(x)=\frac{x^{3}-64}{x^{2}-7 x+12} \] Find the vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two vertical asymptotes. The leftmost asymptote is $\square$ and the rightmost asymptote is (Type equations. Use integers or fractions for any numbers in the equations.) C. The function has no vertical asymptote. Find the horizontal asymptotes. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two horizontal asymptotes. The top asymptote is $\square$ and the rightmost asymptote is (Type equations. Use integers or fractions for any numbers in the equations.) C. The function has no horizontal asymptote. Find the oblique asymptotes. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one oblique asymptote, (Type an equation. Use integers or fractions for any numbers in the equation.) B. The function has two oblique asymptotes. The oblique asymptote with negative slope is $\square$ and the oblique asymptote with positive slope is (Type equations. Use integers or fractions for any numbers in the equations.) C. The function has no oblique asymptote.

Step 1 ：To find the vertical asymptotes of a rational function, we set the denominator equal to zero and solve for x. This is because the function is undefined at these x-values, creating a vertical asymptote.

Step 2 ：For the given function \(H(x)=\frac{x^{3}-64}{x^{2}-7 x+12}\), the denominator is \(x^{2}-7 x+12\). Setting this equal to zero gives us \(x^{2}-7 x+12=0\). Solving this equation gives us the vertical asymptotes at \(x = 3\) and \(x = 4\).

Step 3 ：To find the horizontal asymptotes of a rational function, we compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the x-axis (y = 0) is the horizontal asymptote. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Step 4 ：For the given function \(H(x)=\frac{x^{3}-64}{x^{2}-7 x+12}\), the degree of the numerator is 3 and the degree of the denominator is 2. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Step 5 ：To find the oblique asymptotes of a rational function, we perform long division of the numerator by the denominator. If the degree of the numerator is exactly one more than the degree of the denominator, the quotient is a linear function which is the equation of the oblique asymptote.

Step 6 ：For the given function \(H(x)=\frac{x^{3}-64}{x^{2}-7 x+12}\), the degree of the numerator is 3 and the degree of the denominator is 2. Since the degree of the numerator is exactly one more than the degree of the denominator, there is an oblique asymptote. Performing long division of the numerator by the denominator gives us the oblique asymptote at \(y = x + 7\).

Step 7 ：Final Answer: \(\boxed{\text{The function has two vertical asymptotes at } x = 3 \text{ and } x = 4. \text{ The function has no horizontal asymptote. The function has one oblique asymptote at } y = x + 7.}\)