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Let's analyze the characteristics given for the function \( y = f(x) \):

1. **Limit as \( x \to \infty \)**:  
   \(\lim_{x \to \infty} f(x) = 3\)  
   This implies that as \( x \) approaches infinity, the function \( f(x) \) approaches the horizontal asymptote \( y = 3 \).

2. **Limit as \( x \to -\infty \)**:  
   \(\lim_{x \to -\infty} f(x) = -5\)  
   This suggests that as \( x \) approaches negative infinity, \( f(x) \) approaches the horizontal asymptote \( y = -5 \).

3. **Limit as \( x \to -1^{-} \)**:  
   \(\lim_{x \to -1^{-}} f(x) = \infty\)  
   This means that as \( x \) approaches \(-1\) from the left side, \( f(x) \) goes to positive infinity. This indicates a vertical asymptote at \( x = -1 \), where \( f(x) \to \infty \) from the left.

4. **Limit as \( x \to -1^{+} \)**:  
   \(\lim_{x \to -1^{+}} f(x) = -\infty\)  
   As \( x \) approaches \(-1\) from the right, \( f(x) \) goes to negative infinity. This again confirms a vertical asymptote at \( x = -1 \), where \( f(x) \to -\infty \) from the right.

### Summary of Key Features:
- Horizontal asymptote at \( y = 3 \) as \( x \to \infty \).
- Horizontal asymptote at \( y = -5 \) as \( x \to -\infty \).
- Vertical asymptote at \( x = -1 \), with \( f(x) \to \infty \) from the left and \( f(x) \to -\infty \) from the right.

To identify which graph matches these features, look for:
1. A horizontal line approaching \( y = 3 \) on the far right.
2. A horizontal line approaching \( y = -5 \) on the far left.
3. A vertical line at \( x = -1 \) where the graph goes up to \( +\infty \) on the left side and down to \( -\infty \) on the right side.

Would you like more details or have any further questions?

Here are 5 related questions:
1. What other types of asymptotes can a function have?
2. How does the limit behavior of \( f(x) \) affect the overall shape of the graph?
3. Can a function have multiple vertical asymptotes? If so, how do we identify them?
4. How would the function behave if the limits at infinity were not constants?
5. How does a vertical asymptote differ from a hole in a function's graph?

**Tip**: When analyzing graphs of functions, always pay attention to both horizontal and vertical asymptotes to understand long-term behavior and potential discontinuities.

The graph of \( y = f(x) \) has the following features:

- \( \lim_{x \to \infty} f(x) = 3 \)
- \( \lim_{x \to -\infty} f(x) = -5 \)
- \( \lim_{x \to -1^{-}} f(x) = \infty \)
- \( \lim_{x \to -1^{+}} f(x) = -\infty \)

Which of the following might be a graph of \( y = f(x) \)?

Learn how to identify the graph of \( y = f(x) \) by analyzing limits and asymptotes. This problem explores horizontal and vertical asymptotes and behavior at infinity, suitable for high school calculus students.

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