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For the function g whose graph is given, state the following. (If the answer is positive infinite, type "I"; if negative infinite, type "N"; and if it does not exist, type "D".)

\begin{array}{lll}
(a) \displaystyle\lim_{x \to \infty}g(x)&
(b) \displaystyle\lim_{x \to -\infty}g(x)&
(c) \displaystyle\lim_{x \to 3}g(x)\\
(d) \displaystyle\lim_{x \to 0}g(x)&
(e) \displaystyle\lim_{x \to -2^+}g(x)
\end{array}
(f) The equations of the asymptotes (in increasing order).

(a)
(b)
(c)

(d)
(e)

(f) x=\
, x=\
, and x=\

\ \ y=\
and y=\

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.

For the function g whose graph is given, state the following: (If the answer is positive infinite, type 'I'; if negative infinite, type 'N'; and if it does not exist, type 'D'.)

(a) lim(x → ∞)g(x)
(b) lim(x → -∞)g(x)
(c) lim(x → 3)g(x)
(d) lim(x → 0)g(x)
(e) lim(x → -2+)g(x)
(f) The equations of the asymptotes (in increasing order).

Analyze limits as x approaches infinity, negative infinity, and specific points based on the graph of g(x). This includes identifying vertical and horizontal asymptotes and understanding discontinuities. Suitable for advanced high school and college-level calculus.

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