Question

for the function f whose graph is given, state the following. (a) $limlimits_{x \to infty} f(x)$ $\boxed{-2}$ (checked) (b) $limlimits_{x \to -infty} f(x)$ $\boxed{2}$ (checked) (c) $limlimits_{x \to 1} f(x)$ $\boxed{infty}$ (checked) (d) $limlimits_{x \to 3} f(x)$ $\boxed{-infty}$ (checked) (e) the equations of the asymptotes (enter your answers as a comma-separated list of equations.) vertical $\boxed{}$ × horizontal $\boxed{}$ ×

Explanation:

Part (e) - Vertical Asymptotes

Step1: Analyze vertical asymptote definition

A vertical asymptote occurs where the function approaches \( \pm\infty \). From parts (c) and (d), as \( x \to 1 \), \( f(x) \to \infty \), and as \( x \to 3 \), \( f(x) \to -\infty \). So vertical asymptotes are at \( x = 1 \) and \( x = 3 \).

Step2: Analyze horizontal asymptote definition

A horizontal asymptote is the limit as \( x \to \pm\infty \). From parts (a) and (b), \( \lim_{x\to\infty} f(x) = -2 \) and \( \lim_{x\to-\infty} f(x) = 2 \)? Wait, no, wait—wait, looking at the graph, when \( x \to \infty \), the function approaches a horizontal line, and when \( x \to -\infty \), another? Wait, no, maybe I misread. Wait, part (a) was \( \lim_{x\to\infty} f(x) \), and part (b) \( \lim_{x\to-\infty} f(x) \). Wait, the graph: left end (x→-∞) approaches a horizontal line, right end (x→∞) approaches another? Wait, no, maybe the user's previous answers: part (a) was -2 (correct?), part (b) 2 (correct?). Wait, no, let's re-express.

Wait, vertical asymptotes: where the function has infinite discontinuity. From the graph, there are vertical asymptotes at \( x = 1 \) and \( x = 3 \) (since as x approaches 1, the function goes to infinity, and as x approaches 3, goes to negative infinity).

Horizontal asymptotes: the limit as \( x \to \infty \) and \( x \to -\infty \). From part (a), \( \lim_{x\to\infty} f(x) = -2 \), so horizontal asymptote \( y = -2 \). From part (b), \( \lim_{x\to-\infty} f(x) = 2 \), so horizontal asymptote \( y = 2 \). Wait, but maybe I made a mistake. Wait, the graph: left side (x→-∞) approaches y=2, right side (x→∞) approaches y=-2? Wait, no, the initial problem's part (a) was \( \lim_{x\to\infty} f(x) = -2 \) (marked correct), part (b) \( \lim_{x\to-\infty} f(x) = 2 \) (marked correct). So horizontal asymptotes are \( y = 2 \) (as x→-∞) and \( y = -2 \) (as x→∞). Vertical asymptotes: where the function has vertical breaks, so x=1 and x=3 (since as x approaches 1, f(x)→∞, and x approaches 3, f(x)→-∞).

So vertical asymptotes: \( x = 1 \), \( x = 3 \); horizontal asymptotes: \( y = 2 \), \( y = -2 \).

Answer:

(Vertical Asymptotes):
\( x = 1, x = 3 \)