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The problem requires evaluating several limits based on the given graph of the function \( f(x) = \frac{3}{e^x + 1} \). Let's go through each limit carefully.

### Given:
The function is \( f(x) = \frac{3}{e^x + 1} \).

### Observations from the graph:
- As \( x \to -\infty \), the graph approaches a horizontal asymptote at \( y = 3 \).
- As \( x \to +\infty \), the graph approaches a horizontal asymptote at \( y = 0 \).
- At \( x = 0 \), \( f(x) = 1.5 \).

### Now let's address the limits:

**(a) \( \lim_{x \to -\infty} f(x) \)**  
As \( x \to -\infty \), from the graph, the function value approaches 3.  
So, \( \lim_{x \to -\infty} f(x) = 3 \).

**(b) \( \lim_{x \to +\infty} f(x) \)**  
As \( x \to +\infty \), from the graph, the function approaches 0.  
So, \( \lim_{x \to +\infty} f(x) = 0 \).

**(c) \( \lim_{x \to -1^-} f(x) \)**  
From the graph, as \( x \to -1 \) from the left, the function value approaches approximately 2.  
So, \( \lim_{x \to -1^-} f(x) \approx 2 \).

**(d) \( \lim_{x \to -1^+} f(x) \)**  
As \( x \to -1 \) from the right, the function also approaches approximately 2.  
So, \( \lim_{x \to -1^+} f(x) \approx 2 \).

**(e) \( \lim_{x \to 0} f(x) \)**  
At \( x = 0 \), the graph shows that the function equals 1.5.  
So, \( \lim_{x \to 0} f(x) = 1.5 \).

### Final answers:
- (a) \( \lim_{x \to -\infty} f(x) = 3 \)
- (b) \( \lim_{x \to +\infty} f(x) = 0 \)
- (c) \( \lim_{x \to -1^-} f(x) = 2 \)
- (d) \( \lim_{x \to -1^+} f(x) = 2 \)
- (e) \( \lim_{x \to 0} f(x) = 1.5 \)

Would you like any further details or clarification?

### Related questions:
1. What are horizontal asymptotes, and how do they relate to limits?
2. How do you determine one-sided limits graphically?
3. How do limits behave for rational functions compared to exponential functions?
4. What happens if the limit of a function as \( x \to -\infty \) or \( x \to +\infty \) doesn't exist?
5. How do you handle discontinuities when finding one-sided limits?

### Tip:
When determining limits from graphs, focus on the behavior of the function as it approaches key values (infinity, 0, or discontinuities). This helps you visualize the trend of the function.

Evaluate the given limits using the graph of the function f(x) = 3 / (e^x + 1) and determine the following limits: a) lim_{x -> -∞} f(x), b) lim_{x -> +∞} f(x), c) lim_{x -> -1^-} f(x), d) lim_{x -> -1^+} f(x), e) lim_{x -> 0} f(x).

Explore the process of evaluating limits for the function f(x) = 3 / (e^x + 1) using graph analysis. Understand how limits behave as x approaches infinity and specific values, along with insights on asymptotic behavior and one-sided limits.

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