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

(f(x) = \frac{12x}{4x^2 + 9})

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the horizontal asymptote is (type an equation.)
b. there is no horizontal asymptote.

Question

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

(f(x) = \frac{12x}{4x^2 + 9})

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the horizontal asymptote is   (type an equation.)
b. there is 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": [ "Rational Functions", "Horizontal Asymptotes", "Rational Function End Behavior" ] } ## Identify the degrees of the numerator and denominator Using the *Horizontal Asymptotes* and *Rational Function End Behavior* knowledge points $$ \begin{aligned} &p(x) = 12x \implies \text{degree } n = 1\ &q(x) = 4x^2 + 9 \implies \text{degree } m = 2 \end{aligned} $$ ## Compare the degrees to find the horizontal asymptote Using the *Horizontal Asymptotes* and *Rational Function End Behavior* knowledge points $$ \begin{aligned} &n < m \quad (1 < 2)\ &\lim_{x \to \pm\infty} f(x) = 0 \implies y = 0 \end{aligned} $$

A. The horizontal asymptote is $y = 0$.

B. There is 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: