Question

1. (3pts) let $f(x)$ be a function that satisfies the following.

\\\lim_{x\to\infty} f(x) = \infty\\ \\\lim_{x\to -\infty} f(x) = 6\\ \\\lim_{x\to 3^+} f(x) = \infty\\ \\\lim_{x\to 3^-} f(x) = -\infty\\
$f(x)$ has horizontal asymptote(s) at \underline{\hspace{3cm}} and vertical asymptote(s) at \underline{\hspace{3cm}}

Explanation:

Step1: Identify horizontal asymptotes

A horizontal asymptote exists if $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$ where $L$ is a finite number. Here, $\lim_{x \to -\infty} f(x) = 6$, so this gives a horizontal asymptote. $\lim_{x \to \infty} f(x) = \infty$ does not produce a horizontal asymptote as the limit is infinite.

Step2: Identify vertical asymptotes

A vertical asymptote exists at $x=a$ if $\lim_{x \to a^+} f(x) = \pm\infty$ or $\lim_{x \to a^-} f(x) = \pm\infty$. Here, $\lim_{x \to 3^+} f(x) = \infty$ and $\lim_{x \to 3^-} f(x) = -\infty$, so $x=3$ is a vertical asymptote.

Answer:

$f(x)$ has horizontal asymptote(s) at $\boldsymbol{y=6}$ and vertical asymptote(s) at $\boldsymbol{x=3}$