Find the horizontal asymptote, if any, of the graph of the rational function.

Question

h(x)=10x^3/7x^2+6

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The horizontal asymptote is blank. (Type an equation.)
B. There is no horizontal asymptote.

Answer 1

To find the horizontal asymptote of the rational function \( h(x) = \frac{10x^3}{7x^2 + 6} \), we need to compare the degrees of the polynomial in the numerator and the polynomial in the denominator.

1. The degree of the numerator \( 10x^3 \) is 3.
2. The degree of the denominator \( 7x^2 + 6 \) is 2.

When analyzing horizontal asymptotes for rational functions, we consider the following cases based on the degrees of the numerator (n) and denominator (m):

• If \( n < m \): The horizontal asymptote is \( y = 0 \).
• If \( n = m \): The horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients of the numerator and denominator, respectively.
• If \( n > m \): There is no horizontal asymptote.

In this case, since \( n = 3 \) (numerator) and \( m = 2 \) (denominator), we have \( n > m \). Therefore, there is no horizontal asymptote.

The correct answer is:

B. There is no horizontal asymptote.