Let's analyze the graph and solve each part of the question based on the provided function $g(x)$.

(a) $\lim_{x \to \infty} g(x)$

As $x$ approaches infinity, the graph appears to level off at a specific value, meaning there is a horizontal asymptote. From the graph, it seems that $g(x)$ approaches 0 as $x \to \infty$.

Answer: $0$

(b) $\lim_{x \to -\infty} g(x)$

As $x$ approaches negative infinity, the graph moves downward and appears to tend toward negative infinity.

Answer: $N$ (negative infinity)

(c) $\lim_{x \to 3} g(x)$

As $x$ approaches 3, the graph shows a sharp discontinuity, meaning the limit likely does not exist.

Answer: $D$ (does not exist)

(d) $\lim_{x \to 0} g(x)$

As $x$ approaches 0, the graph smoothly crosses the origin. The value of $g(x)$ at 0 seems to be 0.

Answer: $0$

(e) $\lim_{x \to -2^+} g(x)$

As $x$ approaches $-2$ from the right, the graph seems to tend toward 0.

Answer: $0$

(f) The equations of the asymptotes

The graph shows two types of asymptotes:

• A vertical asymptote occurs around $x = 3$ due to the discontinuity.
• A horizontal asymptote at $y = 0$ as $x \to \infty$.

Answer:

• Vertical asymptote: $x = 3$
• Horizontal asymptote: $y = 0$

Final Answers:

(a) $0$
(b) $N$
(c) $D$
(d) $0$
(e) $0$

(f)

• $x = 3$ (vertical asymptote)
• $y = 0$ (horizontal asymptote)

Do you want any further clarification on this? Here are five related questions you might be interested in:

1. How do you determine the limit of a function from its graph?
2. What is the difference between a horizontal and a vertical asymptote?
3. How can you find limits approaching discontinuities?
4. Can a function have more than one horizontal asymptote?
5. What happens when a function approaches infinity on both ends?

Tip: When analyzing graphs, always check for patterns like asymptotes and discontinuities. This will help in determining limits.