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}\\

Explanation:

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<reasoning>

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

\[

$$\begin{aligned} \lim_{x \to \pm\infty} \frac{3x^2 + 6x - 9}{x^2 - 4} &= \lim_{x \to \pm\infty} \frac{3 + \frac{6}{x} - \frac{9}{x^2}}{1 - \frac{4}{x^2}} \\ y &= 3 \end{aligned}$$

\]
</reasoning>

<answer>
Asíntotas verticales:
\[x = 2, \quad x = -2\]

Asíntota horizontal:
\[y = 3\]
</answer>

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Answer:

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<reasoning>

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

\[

$$\begin{aligned} \lim_{x \to \pm\infty} \frac{3x^2 + 6x - 9}{x^2 - 4} &= \lim_{x \to \pm\infty} \frac{3 + \frac{6}{x} - \frac{9}{x^2}}{1 - \frac{4}{x^2}} \\ y &= 3 \end{aligned}$$

\]
</reasoning>

<answer>
Asíntotas verticales:
\[x = 2, \quad x = -2\]

Asíntota horizontal:
\[y = 3\]
</answer>

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