Question

consider the following rational function. step 2 of 3: find equations for the horizontal or oblique asymptotes, if any, for the function. f(x) = (x² + 10)/(x³ - 64) answerhow to enter your answer (opens in new window) 2 points separate multiple equations with a comma. selecting a button will replace the entered answer value. the value of the button is used instead of the value in the associated box. none keypad keyboard shortcuts

Explanation:

Step1: Analyze degrees of numerator and denominator

For the rational function \( f(x)=\frac{x^{2}+10}{x^{3}-64} \), the degree of the numerator (\( n \)) is 2 (from \( x^{2} \)) and the degree of the denominator (\( d \)) is 3 (from \( x^{3} \)).

Step2: Determine asymptote type

• If \( n < d \), the horizontal asymptote is \( y = 0 \) (x - axis).
• If \( n = d \), the horizontal asymptote is the ratio of leading coefficients.
• If \( n>d \), there is an oblique asymptote (found by long division), but here \( n = 2

Answer:

The horizontal asymptote is \( y = 0 \), and there is no oblique asymptote. So the equation for the horizontal asymptote is \( y = 0 \).