Question

find the horizontal asymptote, if any, of the graph of the rational function.
h(x)=\frac{16x^{3}}{3x^{2}+2}
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the horizontal asymptote is \square. (type an equation.)
b. there is no horizontal asymptote.

Explanation:

Step1: Recall rules for horizontal asymptotes

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

• If the degree of \( f(x) \) (\( n \)) is less than the degree of \( g(x) \) (\( m \)), 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.

Step2: Determine degrees of numerator and denominator

For \( h(x)=\frac{16x^{3}}{3x^{2}+2} \):

• Degree of numerator \( f(x)=16x^{3} \) is \( n = 3 \).
• Degree of denominator \( g(x)=3x^{2}+2 \) is \( m = 2 \).

Step3: Compare degrees

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

Answer:

B. There is no horizontal asymptote.