Find the equation of the horizontal asymptote of the function f left parenthesis x right parenthesis equals Start Fraction x cubed plus 1 over 5 x squared minus 3 x cubed End Fraction.

Question

(1 point)
Responses

y equals five thirds
Image with alt text: y equals five thirds

y equals negative one third
Image with alt text: y equals negative one third

y = 0
y = 0

y equals Start Fraction 1 over 5 End Fraction

Answer 1

To find the horizontal asymptote of the function

\[ f(x) = \frac{x^3 + 1}{5x^2 - 3x^3}, \]

we need to analyze the degrees of the polynomial in the numerator and the denominator.

1. The degree of the numerator \(x^3 + 1\) is 3.
2. The degree of the denominator \(5x^2 - 3x^3\) is also 3 (the highest degree term is \(-3x^3\)).

When the degrees of the numerator and denominator are equal, the horizontal asymptote can be found by taking the ratio of the leading coefficients.

• The leading coefficient in the numerator (from \(x^3\)) is 1.
• The leading coefficient in the denominator (from \(-3x^3\)) is -3.

Thus, the horizontal asymptote is given by

\[ y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{1}{-3} = -\frac{1}{3}. \]

Therefore, the equation of the horizontal asymptote is

\[ \boxed{y = -\frac{1}{3}}. \]