Asymptotes 738C99

1. **Problem Statement:** Determine the equations of all vertical and horizontal asymptotes of the function $f$ and write the equivalent limit expressions. 2. **Recall the definitions:** - A **vertical asymptote** occurs at $x=a$ if $\lim_{x \to a} f(x) = \pm \infty$. - A **horizontal asymptote** occurs at $y=L$ if $\lim_{x \to \pm \infty} f(x) = L$. 3. **Given information:** - Vertical asymptote at $x = -1$ because $\lim_{x \to -1} f(x) = -\infty$. - Vertical asymptote at $x = 5$ because $\lim_{x \to 5} f(x) = \infty$. - Horizontal asymptote at $y = -3$ because $\lim_{x \to \infty} f(x) = -3$. 4. **Equivalent limit equations:** - Vertical asymptotes: $$\lim_{x \to -1} f(x) = -\infty$$ $$\lim_{x \to 5} f(x) = \infty$$ - Horizontal asymptote: $$\lim_{x \to \infty} f(x) = -3$$ 5. **Note:** The graph description mentions the left-hand branch approaches $y=8$ as $x \to -\infty$, but this is not a horizontal asymptote since the problem states the horizontal asymptote is at $y=-3$ for $x \to \infty$. **Final answer:** - Vertical asymptotes: $x = -1$ and $x = 5$ - Horizontal asymptote: $y = -3$ Equivalent limits: $$\lim_{x \to -1} f(x) = -\infty$$ $$\lim_{x \to 5} f(x) = \infty$$ $$\lim_{x \to \infty} f(x) = -3$$