https://r2.erweima.ai/mathbot/image/1726888210112zwm020at.jpg

Let's analyze the graph and solve the limit problems based on the image.

### Given Function: \( g(x) \)

We are asked to solve the following limits:

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**a) \( \lim_{x \to -1} g(x) \)**

Looking at the graph near \( x = -1 \), it appears that as \( x \) approaches \(-1\) from both sides, the function goes to \( -\infty \). Therefore:

\[
\lim_{x \to -1} g(x) = -\infty
\]

---

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

As \( x \to -\infty \), the graph seems to approach a horizontal asymptote at \( y = 2 \).

\[
\lim_{x \to -\infty} g(x) = 2
\]

---

**c) \( \lim_{x \to \infty} g(x) \)**

As \( x \to \infty \), the graph again approaches the same horizontal asymptote at \( y = 2 \).

\[
\lim_{x \to \infty} g(x) = 2
\]

---

**d) \( \lim_{x \to -2} g(x) \)**

Looking at the graph near \( x = -2 \), the function behaves differently from each side:
- As \( x \) approaches \(-2\) from the left, \( g(x) \to -\infty \).
- As \( x \) approaches \(-2\) from the right, \( g(x) \to \infty \).

This indicates a vertical asymptote at \( x = -2 \), so:

\[
\lim_{x \to -2} g(x) \text{ does not exist (DNE)}
\]

---

**e) \( \lim_{x \to -1^+} g(x) \)**

This is the right-hand limit at \( x = -1 \). As \( x \) approaches \(-1\) from the right, the graph goes to \( -\infty \).

\[
\lim_{x \to -1^+} g(x) = -\infty
\]

---

**f) \( \lim_{x \to -3} g(x) \)**

At \( x = -3 \), the function appears to have a finite value. From the graph, we see that:

\[
g(-3) = 0
\]

Thus:

\[
\lim_{x \to -3} g(x) = 0
\]

---

### Summary of Limits:
a) \( \lim_{x \to -1} g(x) = -\infty \)  
b) \( \lim_{x \to -\infty} g(x) = 2 \)  
c) \( \lim_{x \to \infty} g(x) = 2 \)  
d) \( \lim_{x \to -2} g(x) \text{ DNE} \)  
e) \( \lim_{x \to -1^+} g(x) = -\infty \)  
f) \( \lim_{x \to -3} g(x) = 0 \)  

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Would you like further details or clarifications on any of the limits?

### Here are five related questions for deeper understanding:
1. What is the significance of a vertical asymptote in limits?
2. How can you determine if a limit at infinity represents a horizontal asymptote?
3. Why does the limit at \( x = -2 \) not exist?
4. What are the left-hand and right-hand limits, and how do they relate to overall limits?
5. How would the limits change if the function had different asymptotes?

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### Tip:
When approaching limits involving infinity, always consider the behavior of the graph as it extends to very large or small values of \( x \). This helps identify horizontal asymptotes.

Determine the following limits based on the graph of the function g(x): a) lim(x→−1) g(x), b) lim(x→−∞) g(x), c) lim(x→∞) g(x), d) lim(x→−2) g(x), e) lim(x→−1+) g(x), f) lim(x→−3) g(x)

This problem involves determining various limits of the function g(x) as x approaches specific values. Using the graph, students can find limits such as lim(x→−1) g(x) = -∞, lim(x→−∞) g(x) = 2, and lim(x→−3) g(x) = 0. Learn about vertical asymptotes and behavior at infinity.

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