Find the Asymptotes f(x)=(2x^3-3x-9)/(9x^3-5x^2+3)

Problem

ƒ(x)=(2*x3−3*x−9)/(9*x3−5*x2+3)

Solution

1. Identify the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since both the numerator and the denominator are polynomials of degree 3, the horizontal asymptote is the ratio of the leading coefficients.

2. Calculate the ratio of the leading coefficients. The leading coefficient of the numerator is 2 and the leading coefficient of the denominator is 9

y=2/9

3. Identify the vertical asymptotes by finding the values of x that make the denominator equal to zero. We set the denominator to zero.

9*x3−5*x2+3=0

4. Solve for x using numerical methods or the cubic formula, as the denominator does not factor easily. Testing for real roots shows there is one real root.

x≈−0.585

5. Check for slant asymptotes. Since the degree of the numerator is not exactly one greater than the degree of the denominator, there is no slant (oblique) asymptote.

Final Answer

Horizontal Asymptote: *y=2/9, Vertical Asymptote: *x≈−0.585

Want more problems? Check here!