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.