To find the horizontal asymptote of the rational function

$h(x) = \frac{10x^3}{7x^2 + 8},$

we need to compare the degrees of the polynomial in the numerator and the polynomial in the denominator.

1. Degree of the Numerator: The highest power of $x$ in the numerator is $x^3$, so the degree of the numerator is 3.
2. Degree of the Denominator: The highest power of $x$ in the denominator is $x^2$, so the degree of the denominator is 2.

Rule for Horizontal Asymptotes:

• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, the function has an oblique (or slant) asymptote.

Since the degree of the numerator (3) is greater than the degree of the denominator (2), this function does not have a horizontal asymptote.

Answer:

The correct choice is: B. There is no horizontal asymptote.

Would you like more details on horizontal asymptotes or have any further questions?

Here are some related questions to deepen your understanding:

1. What is an oblique asymptote, and how is it calculated?
2. How would the answer change if the degrees of the numerator and denominator were the same?
3. What happens to the horizontal asymptote if the degree of the denominator is greater than the degree of the numerator?
4. How can you determine the behavior of $h(x)$ as $x \to \infty$?
5. Why does the presence of a higher-degree term in the numerator affect the asymptote?

Tip: Always compare the degrees of the numerator and denominator first when looking for horizontal asymptotes in rational functions.