Question

find the horizontal asymptote, if any, of the graph of the rational function.
$h(x) = \frac{12x^3}{4x^2 + 7}$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the horizontal asymptote is \boxed{}. (type an equation.)

b. there is no horizontal asymptote.

Explanation:

Step1: Recall Horizontal Asymptote Rules

For a rational function \( h(x)=\frac{f(x)}{g(x)} \), where \( f(x) \) and \( g(x) \) are polynomials:

• If the degree of \( f(x) \) (let \( n \)) is less than the degree of \( g(x) \) (let \( m \)), horizontal asymptote is \( y = 0 \).
• If \( n = m \), horizontal asymptote is \( y=\frac{\text{leading coefficient of }f(x)}{\text{leading coefficient of }g(x)} \).
• If \( n>m \), there is no horizontal asymptote (but there may be an oblique asymptote).

Step2: Determine Degrees of Numerator and Denominator

For \( h(x)=\frac{12x^{3}}{4x^{2}+7} \):

• Degree of numerator \( f(x)=12x^{3} \) is \( n = 3 \).
• Degree of denominator \( g(x)=4x^{2}+7 \) is \( m = 2 \).

Since \( n = 3>m = 2 \), by the rules, there is no horizontal asymptote.

Answer:

B. There is no horizontal asymptote.