Question

find the horizontal asymptote, if any, of the graph of the rational function

h(x)=\frac{10 x^{3}}{7 x^{2}+3}

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

a. the horizontal asymptote is . (type an equation.)
b. there is no horizontal asymptote.

Explanation:

Step1: Analyze the degrees of numerator and denominator

For the rational function \( h(x)=\frac{10x^{3}}{7x^{2}+3} \), the degree of the numerator \( n = 3 \) (since the highest - power of \( x \) in the numerator is \( x^{3} \)) and the degree of the denominator \( m=2 \) (since the highest - power of \( x \) in the denominator is \( x^{2} \)).

Step2: Apply the horizontal asymptote rule

The rule for horizontal asymptotes of a rational function \( y = \frac{f(x)}{g(x)}=\frac{a_nx^{n}+a_{n - 1}x^{n-1}+\cdots+a_0}{b_mx^{m}+b_{m - 1}x^{m-1}+\cdots+b_0} \) is:
If \( n>m \), then there is no horizontal asymptote.

Since \( n = 3\) and \( m = 2\) (\(n>m\)), we use the rule for horizontal asymptotes of rational functions.

Answer:

B. There is no horizontal asymptote.