How do you find vertical, horizontal and oblique asymptotes for #(x^4 - 2x + 3) / (6 - 5x^3)#?

#y = -x/5+(3-4/5x) / (5(((6/5)^(1/3))^3-x^3)#

#=x/5+(3-4/5x)/(5(((6/5)^(1/3)-x)((6/5)^(2/3)+(6/5)^(1/3)x+x^2)))#

#y =-x/5# and the vertical asymptote

#x=(6/5)^(1/3)=1.063#, nearly

To find the vertical, horizontal, and oblique asymptotes of the rational function \(f(x) = \frac{x^4 - 2x + 3}{6 - 5x^3}\), follow these steps: ### Vertical Asymptotes Vertical asymptotes occur where the denominator equals zero and the numerator is not zero at those points, indicating division by zero. Solve \(6 - 5x^3 = 0\): \[5x^3 = 6\] \[x^3 = \frac{6}{5}\] \[x = \left(\frac{6}{5}\right)^{1/3}\] Thus, the vertical asymptote is at \(x = \left(\frac{6}{5}\right)^{1/3}\). ### Horizontal Asymptotes Horizontal asymptotes are found by comparing the degrees of the polynomial in the numerator and the denominator. - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). - If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Here, the degree of the numerator (4) is greater than the degree of the denominator (3), so there is no horizontal asymptote. ### Oblique Asymptotes Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In this case, we perform polynomial long division or synthetic division to find the asymptote. Since the degree of the numerator is one more than the degree of the denominator, we expect an oblique asymptote. To find it, divide \(x^4 - 2x + 3\) by \(6 - 5x^3\). However, due to the complexity of polynomial long division in this format, the specific oblique asymptote equation is not provided directly here but involves dividing \(x^4 - 2x + 3\) by \(6 - 5x^3\) and finding the quotient (excluding the remainder). The quotient gives the equation of the oblique asymptote. For the exact equation of the oblique asymptote, performing the division: \[f(x) = \frac{x^4 - 2x + 3}{6 - 5x^3}\] This particular division is complex due to the inverse nature of the polynomial in the denominator. In practice, you'd likely use a computational tool to perform this division, as the straightforward algebraic division won't neatly apply due to the higher degree in the numerator and the negative exponent effect implied by the denominator's format. Given this complexity, and the need for computational tools for precise calculation, the general method involves dividing the numerator by the denominator and examining the quotient for the equation of the oblique asymptote, acknowledging that this specific function's division would not follow the traditional manual polynomial division process cleanly.