https://file.aiquickdraw.com/mathbot/file/17376111113300qwh2dcw.jpg

limx→5^- limx→5^+ limx→3 limx→infty limx→- infty

The graph you provided appears to represent a function with significant changes near specific points, particularly at \( x = 5 \) and potentially at \( x = 3 \). Let's analyze the following limits based on the graph.

1. **Limit as \( x \to 5^- \) (left-hand limit at \( x = 5 \)):**
   The graph approaches positive infinity as \( x \) gets close to 5 from the left side. Hence, the limit is \( \lim_{x \to 5^-} f(x) = +\infty \).

2. **Limit as \( x \to 5^+ \) (right-hand limit at \( x = 5 \)):**
   Similarly, the graph approaches negative infinity as \( x \) approaches 5 from the right side. Hence, the limit is \( \lim_{x \to 5^+} f(x) = -\infty \).

3. **Limit as \( x \to 3 \):**
   The function seems to have some behavior near \( x = 3 \), but the graph doesn't indicate a specific value, potentially suggesting a discontinuity or undefined limit at \( x = 3 \). The limit may be either undefined or could approach infinity.

4. **Limit as \( x \to \infty \):**
   As \( x \) increases, the graph seems to approach a horizontal line, which implies the function has a horizontal asymptote. From the graph, it appears the limit as \( x \to \infty \) is \( y = 0 \), or \( \lim_{x \to \infty} f(x) = 0 \).

5. **Limit as \( x \to -\infty \):**
   Similarly, as \( x \to -\infty \), the graph appears to approach \( y = 0 \), so \( \lim_{x \to -\infty} f(x) = 0 \).

Would you like further analysis on specific sections of the graph, or do you have any particular questions regarding these limits?

### Related Questions:
1. How do we determine the left-hand and right-hand limits graphically?
2. What happens when the limit from one side does not match the limit from the other side?
3. How do horizontal asymptotes relate to limits at infinity?
4. What is the significance of a discontinuity in a function?
5. How can we find limits of functions that approach infinity at certain points?

**Tip:** Always check both the left-hand and right-hand limits when dealing with discontinuities or undefined behavior at certain points!

limx→5^- limx→5^+ limx→3 limx→infty limx→-infty

Explore the graphical analysis of limits, including left-hand and right-hand limits, behavior at infinity, and the significance of horizontal asymptotes. Learn how to interpret a function's graph and identify limits at critical points like x=5 and x=3.

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