find the horizontal asymptote, if any, of the graph of the rational function.

(f(x) = \frac{-6x}{3x^3 + x^2 + 1})

a. (y = 0)
b. (y = -\frac{1}{2})
c. (y = -2)
d. no horizontal asymptote

Question

find the horizontal asymptote, if any, of the graph of the rational function.

(f(x) = \frac{-6x}{3x^3 + x^2 + 1})

a. (y = 0)
b. (y = -\frac{1}{2})
c. (y = -2)
d. no horizontal asymptote

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Horizontal Asymptotes", "Rational Function End Behavior" ], "new_concepts": [], "current_concepts": [ "Horizontal Asymptotes", "Rational Function End Behavior", "Degree of Polynomial" ] } ## Identify the degrees of the numerator and denominator Using the *Horizontal Asymptotes* and *Rational Function End Behavior* knowledge points $$ \begin{aligned} &p(x) = -6x \implies \text{degree } n = 1\ &q(x) = 3x^3 + x^2 + 1 \implies \text{degree } m = 3 \end{aligned} $$ ## Compare the degrees to find the horizontal asymptote Using the *Horizontal Asymptotes* and *Rational Function End Behavior* knowledge points $$ \begin{aligned} &n < m \quad (1 < 3)\ &\lim_{x \to \pm\infty} f(x) = 0 \implies y = 0 \end{aligned} $$

(A) $y = 0$

(B) $y = -\frac{1}{2}$

(C) $y = -2$

(D) no horizontal asymptote

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Horizontal Asymptotes" ] }

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QUESTION IMAGE

Question

Explanation:

Identify the degrees of the numerator and denominator

Compare the degrees to find the horizontal asymptote

Answer: