hallar las asíntotas de una función racional: avanzado

trazar todas las asíntotas verticales y horizontales de la función.

\f(x) = \frac{3x^2 + 6x - 9}{x^2 - 4}\

Question

hallar las asíntotas de una función racional: avanzado

trazar todas las asíntotas verticales y horizontales de la función.

\f(x) = \frac{3x^2 + 6x - 9}{x^2 - 4}\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes", "Rational Functions" ], "current_concepts": [ "Vertical Asymptotes", "Horizontal Asymptotes", "Rational Functions" ] } ## Identificar las asíntotas verticales $$ \begin{aligned} x^2 - 4 &= 0 \ (x - 2)(x + 2) &= 0 \ x = 2, \quad x &= -2 \end{aligned} $$ ## Identificar la asíntota horizontal $$ \begin{aligned} \lim_{x \to \pm\infty} \frac{3x^2 + 6x - 9}{x^2 - 4} &= \lim_{x \to \pm\infty} \frac{3 + \frac{6}{x} - \frac{9}{x^2}}{1 - \frac{4}{x^2}} \ y &= 3 \end{aligned} $$ Asíntotas verticales: $$x = 2, \quad x = -2$$ Asíntota horizontal: $$y = 3$$ { "elements": [ { "type": "line", "params": [[2, -10], [2, 10]], "properties": { "name": "x = 2", "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "line", "params": [[-2, -10], [-2, 10]], "properties": { "name": "x = -2", "strokeColor": "#F2557F", "strokeWidth": 2, "dash": 2, "withLabel": true } }, { "type": "line", "params": [[-10, 3], [10, 3]], "properties": { "name": "y = 3", "strokeColor": "#8C55F2", "strokeWidth": 2, "dash": 2, "withLabel": true } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Rational Functions" ] }

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Question

Explanation:

Identificar las asíntotas verticales

Identificar la asíntota horizontal

Answer: