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.

Explanation:

<pre_analysis>
{
"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"
]
}
</pre_analysis>

<reasoning>

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
\[

$$\begin{aligned} &n < m \quad (1 < 2)\\ &\lim_{x \to \pm\infty} f(x) = 0 \implies y = 0 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>A. The horizontal asymptote is \(y = 0\).</mcq-correct>
<mcq-option>B. There is no horizontal asymptote.</mcq-option>
</answer>

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

Answer:

<pre_analysis>
{
"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"
]
}
</pre_analysis>

<reasoning>

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
\[

$$\begin{aligned} &n < m \quad (1 < 2)\\ &\lim_{x \to \pm\infty} f(x) = 0 \implies y = 0 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-correct>A. The horizontal asymptote is \(y = 0\).</mcq-correct>
<mcq-option>B. There is no horizontal asymptote.</mcq-option>
</answer>

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