Let's go over each part of the question based on the graph of the function $f(x)$.

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

As $x \to \infty$, the graph shows that the function approaches the horizontal line at $y = 1$. Therefore, the limit is: $\lim_{x \to \infty} f(x) = 1$

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

As $x \to -\infty$, the function tends towards the horizontal line at $y = -1$. Hence, the limit is: $\lim_{x \to -\infty} f(x) = -1$

(c) $\lim_{x \to 1} f(x)$

From the graph, the function has a peak at $x = 1$, but there is no discontinuity. The function approaches $f(1) = 3$ from both sides as $x \to 1$. Thus, the limit is: $\lim_{x \to 1} f(x) = 3$

(d) $\lim_{x \to 3} f(x)$

At $x = 3$, there is a vertical asymptote. As $x \to 3$, the function tends to $-\infty$. Hence, the limit is: $\lim_{x \to 3} f(x) = -\infty$

Summary of Limits:

• $\lim_{x \to \infty} f(x) = 1$
• $\lim_{x \to -\infty} f(x) = -1$
• $\lim_{x \to 1} f(x) = 3$
• $\lim_{x \to 3} f(x) = -\infty$

Would you like any further details on these limits or have additional questions?

Here are five related questions:

1. What is a horizontal asymptote, and how does it relate to limits at infinity?
2. How can we determine vertical asymptotes from a graph?
3. What happens if the limit from the left does not equal the limit from the right at a specific point?
4. Can a function have different limits at infinity for positive and negative directions?
5. What is the significance of a function approaching infinity or negative infinity near a vertical asymptote?

Tip: When analyzing a graph for limits, check for asymptotic behavior and continuity at key points. This helps in determining limits, especially near infinity and potential discontinuities.