Let's analyze the limits and asymptotes based on the graph provided.

Solutions:

(a) $\lim_{x \to \infty} g(x)$: As $x$ approaches infinity, the graph appears to level off at a particular value. Observing the graph, it seems to approach a horizontal asymptote around $y = 0$, so: $\lim_{x \to \infty} g(x) = 0$

(b) $\lim_{x \to -\infty} g(x)$: As $x$ approaches negative infinity, the graph oscillates but stabilizes near $y = 0$. Thus: $\lim_{x \to -\infty} g(x) = 0$

(c) $\lim_{x \to 0} g(x)$: Looking closely at $x = 0$, the graph does not show any significant oscillations or divergences around this point, so the function appears to approach a specific value. If the graph passes through $y = 1$ at $x = 0$, then: $\lim_{x \to 0} g(x) = 1$

(d) $\lim_{x \to 2^-} g(x)$: As $x$ approaches $2$ from the left, the graph heads towards positive infinity, indicating a vertical asymptote: $\lim_{x \to 2^-} g(x) = \infty$

(e) $\lim_{x \to 2^+} g(x)$: As $x$ approaches $2$ from the right, the graph goes towards negative infinity, reinforcing the presence of a vertical asymptote at $x = 2$: $\lim_{x \to 2^+} g(x) = -\infty$

(f) Equations of the asymptotes:

• Vertical asymptote: Based on the behavior around $x = 2$, there is a vertical asymptote at $x = 2$.
• Horizontal asymptote: As $x$ approaches both positive and negative infinity, $y$ approaches 0, so there is a horizontal asymptote at $y = 0$.

Thus, the asymptote equations are: $x = 2, \quad y = 0$

Would you like further details on any of these solutions?

Related Questions

1. What is the significance of the horizontal asymptote in this function?
2. How would the behavior change if the graph had an oscillating limit at infinity?
3. Why does the function exhibit a vertical asymptote at $x = 2$?
4. How do horizontal asymptotes influence the end behavior of functions?
5. Can you determine the continuity of $g(x)$ based on the given limits?

Tip:

When analyzing a graph for asymptotes, always check both directions for horizontal asymptotes and both sides of points with potential vertical asymptotes to confirm the function’s behavior near those points.