Determine if the function below has a horizontal or a slant asymptote, and explain why. \[ f(x)=\frac{-11 x-23 x^{2}-6-12 x^{3}}{2+x+4 x^{2}} \] Answer Attempt 1 out of 2 The function has a \( \quad \vee \) asymptote because the degree of the numerator is \( \square \) the degree of the denominator. As \( x \) approaches infinity (positive or negative), the value of \( y \) approaches \( \square \).

To determine if the function \( f(x) = \frac{-11x - 23x^2 - 6 - 12x^3}{2 + x + 4x^2} \) has a horizontal or slant asymptote, we need to analyze the degrees of the numerator and the denominator. 1. **Identify the degrees of the numerator and denominator:** - The numerator is \( -11x - 23x^2 - 6 - 12x^3 \). The highest degree term is \( -12x^3 \), so the degree of the numerator \( n = 3 \). - The denominator is \( 2 + x + 4x^2 \). The highest degree term is \( 4x^2 \), so the degree of the denominator \( m = 2 \). 2. **Compare the degrees:** - Since the degree of the numerator \( n = 3 \) is greater than the degree of the denominator \( m = 2 \) (i.e., \( n > m \)), the function does not have a horizontal asymptote. Instead, it may have a slant (oblique) asymptote. 3. **Finding the slant asymptote:** - To find the slant asymptote, we perform polynomial long division of the numerator by the denominator. Let's perform the polynomial long division to find the slant asymptote. ### Polynomial Long Division 1. Divide the leading term of the numerator by the leading term of the denominator: \[ \frac{-12x^3}{4x^2} = -3x \] 2. Multiply the entire denominator by \(-3x\): \[ -3x(2 + x + 4x^2) = -6x - 3x^2 - 12x^3 \] 3. Subtract this from the original numerator: \[ (-11x - 23x^2 - 6 - 12x^3) - (-12x^3 - 3x^2 - 6x) = (-11x + 6x) + (-23x^2 + 3x^2) - 6 = -5x - 20x^2 - 6 \] 4. Now, divide the new leading term by the leading term of the denominator: \[ \frac{-20x^2}{4x^2} = -5 \] 5. Multiply the entire denominator by \(-5\): \[ -5(2 + x + 4x^2) = -10 - 5x - 20x^2 \] 6. Subtract this from the previous result: \[ (-5x - 20x^2 - 6) - (-20x^2 - 5x - 10) = -6 + 10 = 4 \] The result of the division is: \[ f(x) = -3x - 5 + \frac{4}{2 + x + 4x^2} \] ### Conclusion As \( x \) approaches infinity (positive or negative), the term \( \frac{4}{2 + x + 4x^2} \) approaches \( 0 \). Therefore, the slant asymptote is given by: \[ y = -3x - 5 \] ### Final Answer The function has a slant asymptote because the degree of the numerator is greater than the degree of the denominator. As \( x \) approaches infinity (positive or negative), the value of \( y \) approaches \( -3x - 5 \).