for what values of a and m does f(x) have a horizontal asymptote at y = 2 and a vertical asymptote at x = 1?

(f(x) = \frac{2x^m}{x+a})

$a = -1, m = 0$
$a = 1, m = 0$
$a = -1, m = 1$
$a = 1, m = 1$

Question

for what values of a and m does f(x) have a horizontal asymptote at y = 2 and a vertical asymptote at x = 1?

(f(x) = \frac{2x^m}{x+a})

$a = -1, m = 0$
$a = 1, m = 0$
$a = -1, m = 1$
$a = 1, m = 1$

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": [ "Vertical Asymptotes", "Horizontal Asymptotes" ], "new_concepts": [], "current_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes", "Rational Functions" ] } ## Analyze the vertical asymptote Using the *Vertical Asymptotes* knowledge point $$ \begin{aligned} &x + a = 0 \quad \text{at } x = 1 \ &1 + a = 0 \ &a = -1 \end{aligned} $$ ## Analyze the horizontal asymptote Using the *Horizontal Asymptotes* knowledge point $$ \begin{aligned} &f(x) = \frac{2x^m}{x + a} \ &\text{Since } y = 2 \text{ is a non-zero horizontal asymptote, the degrees must be equal:} \ &m = 1 \ &\lim_{x \to \infty} \frac{2x^1}{x - 1} = 2 \end{aligned} $$ ## Match the parameters to the options Using the *Rational Functions* knowledge point $$ \begin{aligned} &a = -1 \ &m = 1 \end{aligned} $$

(A) $a = -1, m = 0$

(B) $a = 1, m = 0$

(C) $a = -1, m = 1$

(D) $a = 1, m = 1$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Asymptotes of Rational Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the vertical asymptote

Analyze the horizontal asymptote

Match the parameters to the options

Answer: