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what’s lim x approaches 3 minus and lim x approaches 3 plus and lim x approaches 3

Looking at the graph you've provided, I can see that we need to find the limits of the function as \( x \) approaches 3 from the left (denoted as \( \lim_{x \to 3^-} \)), from the right (denoted as \( \lim_{x \to 3^+} \)), and the overall limit as \( x \) approaches 3 (denoted as \( \lim_{x \to 3} \)).

Let's break it down:

1. **\( \lim_{x \to 3^-} \)**: This is the limit as \( x \) approaches 3 from the left side (values less than 3). 
   - As \( x \) approaches 3 from the left, the y-value is approaching 4.

2. **\( \lim_{x \to 3^+} \)**: This is the limit as \( x \) approaches 3 from the right side (values greater than 3).
   - As \( x \) approaches 3 from the right, the y-value is also approaching 4.

3. **\( \lim_{x \to 3} \)**: This is the overall limit, which exists only if both the left-hand limit and the right-hand limit are the same.
   - Since \( \lim_{x \to 3^-} = \lim_{x \to 3^+} = 4 \), we conclude that \( \lim_{x \to 3} = 4 \).

### Summary:
- \( \lim_{x \to 3^-} = 4 \)
- \( \lim_{x \to 3^+} = 4 \)
- \( \lim_{x \to 3} = 4 \)

Let me know if you would like more details or if you have any questions!

Here are 5 related questions for deeper understanding:
1. What is the definition of a limit from the left and from the right?
2. How do we determine if the limit at a point exists?
3. What happens if the left-hand and right-hand limits are not equal?
4. Can a function have a limit at a point if it is not continuous at that point?
5. How does the concept of limits relate to derivatives in calculus?

**Tip:** When graphing limits, always check both the left and right approaches to ensure consistency!

What’s lim x approaches 3 minus, lim x approaches 3 plus, and lim x approaches 3?

Analyze the limits of a function as x approaches 3 using a graph. Determine the left-hand limit, right-hand limit, and overall limit of the function based on the values the function approaches from both sides. Ideal for students learning calculus concepts of limits.

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