https://file.aiquickdraw.com/mathbot/file/172645029813257oinulz.jpg

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?

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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.

For the function f whose graph is given, state the following limits: lim x→∞ f(x), lim x→-∞ f(x), lim x→1 f(x), and lim x→3 f(x).

Learn how to evaluate limits and identify asymptotes from the graph of a function. This problem covers the analysis of horizontal and vertical asymptotic behavior, including limits as x approaches infinity and at critical points of discontinuity.

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