find the horizontal asymptote of ( f(x)=\frac{5 x+3 x^{3}+1}{-5 x^{3}-2 x^{2}-3} ). if the horizontal…

find the horizontal asymptote of ( f(x)=\frac{5 x+3 x^{3}+1}{-5 x^{3}-2 x^{2}-3} ). if the horizontal asymptote does not exist, enter dne. the horizontal asymptote is ( y= )

find the horizontal asymptote of ( f(x)=\frac{5 x+3 x^{3}+1}{-5 x^{3}-2 x^{2}-3} ). if the horizontal asymptote does not exist, enter dne. the horizontal asymptote is ( y= )

Answer

Explanation:

Step1: Identify the degrees of numerator and denominator

The degree of the numerator (5x + 3x^{3}+1) (highest - power term is (3x^{3})) is (n = 3). The degree of the denominator (-5x^{3}-2x^{2}-3) (highest - power term is (-5x^{3})) is (m = 3).

Step2: Use the horizontal - asymptote rule for rational functions

When (n=m), the horizontal asymptote (y=\frac{a_{n}}{b_{m}}), where (a_{n}) is the leading coefficient of the numerator and (b_{m}) is the leading coefficient of the denominator. For the function (f(x)=\frac{5x + 3x^{3}+1}{-5x^{3}-2x^{2}-3}), (a_{n}=3) and (b_{m}=-5). So, (y = \frac{3}{-5}=-\frac{3}{5})

Answer:

(-\frac{3}{5})