To find the horizontal asymptote of the function

$f(x) = \frac{3x^5 - 8x^3}{4x^4 + 7x^3}$

we follow these steps:

1. Compare the Degrees:

   • The degree of the numerator is $5$ (highest power of $x$ in $3x^5$).
   • The degree of the denominator is $4$ (highest power of $x$ in $4x^4$).

2. Determine Asymptote:

   • If the degree of the numerator is greater than the degree of the denominator, the function has no horizontal asymptote. Instead, there is a slant (or oblique) asymptote.
   • Here, the degree of the numerator (5) is greater than the degree of the denominator (4). Therefore, there is no horizontal asymptote.

Would you like further details or have any questions?

Related Questions

1. What is the difference between a horizontal and a slant asymptote?
2. How do you find the slant asymptote for this function?
3. What are the vertical asymptotes, if any, for this function?
4. How would the asymptotes change if the degrees of the numerator and denominator were equal?
5. What does the behavior of the function look like as $x \to \infty$ or $x \to -\infty$?

Tip

Remember, when determining asymptotes, always start by comparing the degrees of the numerator and the denominator for quick insights!