Find the Horizontal Asymptote of this function... \[ \begin{array}{l}\frac{5 x^{3}-x^{2}+6}{x-4}\end{array} \]

To determine the horizontal asymptote of the function: \[ f(x) = \frac{5x^3 - x^2 + 6}{x - 4} \] we analyze the degrees of the polynomial in the numerator and the denominator. 1. **Degree of the Numerator (P(x))**: The highest power of \( x \) in the numerator is \( x^3 \), so the degree is **3**. 2. **Degree of the Denominator (Q(x))**: The highest power of \( x \) in the denominator is \( x \), so the degree is **1**. **Rules for Horizontal Asymptotes:** - **If the degree of \( P(x) \) < degree of \( Q(x) \):** The horizontal asymptote is \( y = 0 \). - **If the degree of \( P(x) \) = degree of \( Q(x) \):** The horizontal asymptote is \( y = \frac{\text{Leading Coefficient of } P(x)}{\text{Leading Coefficient of } Q(x)} \). - **If the degree of \( P(x) \) > degree of \( Q(x) \):** There is **no horizontal asymptote**. Instead, the function may have an oblique (slant) asymptote or a more complex behavior. In this case, since the degree of the numerator (3) is greater than the degree of the denominator (1), **there is no horizontal asymptote** for the function \( f(x) \). **Conclusion:** The function \( \frac{5x^3 - x^2 + 6}{x - 4} \) does **not** have a horizontal asymptote. **Answer:** There is no horizontal asymptote for the given function.