what are the vertical and horizontal asymptotes for the function $f(x) = \frac{3x^2}{x^2 - 4}$?

horizontal asymptote: $y = -2, y = 2$
vertical asymptote: $x = 3$

horizontal asymptote: $y = -4, y = 1$
vertical asymptote: $x = 3$

horizontal asymptote: $y = 3$
vertical asymptote: $x = -4, x = 1$

horizontal asymptote: $y = 3$
vertical asymptote: $x = -2, x = 2$

Question

what are the vertical and horizontal asymptotes for the function $f(x) = \frac{3x^2}{x^2 - 4}$?

horizontal asymptote: $y = -2, y = 2$
vertical asymptote: $x = 3$

horizontal asymptote: $y = -4, y = 1$
vertical asymptote: $x = 3$

horizontal asymptote: $y = 3$
vertical asymptote: $x = -4, x = 1$

horizontal asymptote: $y = 3$
vertical asymptote: $x = -2, x = 2$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [], "current_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes", "Rational Functions" ] } ## Find the vertical asymptotes $$ \begin{aligned} x^2 - 4 &= 0 \ (x - 2)(x + 2) &= 0 \ x = -2, \quad &x = 2 \end{aligned} $$ ## Find the horizontal asymptote $$ \begin{aligned} \lim_{x \to \pm\infty} \frac{3x^2}{x^2 - 4} &= \lim_{x \to \pm\infty} \frac{3}{1 - \frac{4}{x^2}} \ &= 3 \ y &= 3 \end{aligned} $$ ## Match the correct option $$ \begin{aligned} \text{horizontal asymptote: } &y = 3 \ \text{vertical asymptote: } &x = -2, \ x = 2 \end{aligned} $$

horizontal asymptote: $y = -2, y = 2$
vertical asymptote: $x = 3$

horizontal asymptote: $y = -4, y = 1$
vertical asymptote: $x = 3$

horizontal asymptote: $y = 3$
vertical asymptote: $x = -4, x = 1$

horizontal asymptote: $y = 3$
vertical asymptote: $x = -2, x = 2$

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find the vertical asymptotes

Find the horizontal asymptote

Match the correct option

Answer: