find the horizontal asymptote of ( f(x)=\frac{-2 x-5 x^{3}+4}{3 x^{3}-3 x^{2}-1} ).\n( y= )
find the horizontal asymptote of ( f(x)=\frac{-2 x-5 x^{3}+4}{3 x^{3}-3 x^{2}-1} ).\n( y= )
Answer
Explanation:
Step1: Determine the degrees of numerator and denominator
The degree of the numerator ( -2x - 5x^{3}+4) is (n = 3) (since the highest - power term is (-5x^{3})), and the degree of the denominator (3x^{3}-3x^{2}-1) is (m = 3).
Step2: Use the horizontal asymptote rule for (n = m)
When (n=m), the horizontal asymptote (y) is given by the ratio of the leading coefficients. The leading coefficient of the numerator (a=-5) (from (-5x^{3})) and the leading coefficient of the denominator (b = 3) (from (3x^{3})). So, (y=\frac{a}{b}).
Answer:
(y =-\frac{5}{3})