From the uploaded image, the graph appears to depict a rational function with two horizontal asymptotes and a vertical asymptote at $x = -2$. Let's analyze the graph systematically:

Observations:

1. Horizontal asymptotes:

   • As $x \to -\infty$, $f(x) \to -1$ (function approaches the horizontal asymptote from below).
   • As $x \to \infty$, $f(x) \to -1$ (function approaches the horizontal asymptote from below).

2. Vertical asymptote:

   • At $x = -2$:
     • As $x \to -2^-$, $f(x) \to \infty$ (approaches positive infinity from the left).
     • As $x \to -2^+$, $f(x) \to -\infty$ (approaches negative infinity from the right).

Correct Answers:

Based on the graph:

• Correct:

  • As $x \to -\infty$, $f(x) \to -1$.
  • As $x \to \infty$, $f(x) \to -1$.
  • As $x \to -2^-$, $f(x) \to \infty$.

• Incorrect:

  • As $x \to -2^-$, $f(x) \to -\infty$ (this occurs for $x \to -2^+$, not $x \to -2^-$).
  • As $x \to -\infty$, $f(x) \to 1$.
  • As $x \to \infty$, $f(x) \to 1$.

Would you like further details on how to analyze these asymptotes or explore rational function behaviors?

Related Questions:

1. What determines the horizontal asymptote of a rational function?
2. How do vertical asymptotes influence the behavior of a function near certain points?
3. Can a function have multiple horizontal asymptotes? Why or why not?
4. What mathematical techniques are used to confirm the limits at infinity analytically?
5. How does the degree of the numerator and denominator influence asymptotic behavior?

Tip: Horizontal asymptotes indicate end behavior as $x \to \pm \infty$, while vertical asymptotes describe unbounded behavior near specific $x$-values.