https://file.aiquickdraw.com/mathbot/file/1726705313000m3n6v641.jpg

From the graph you uploaded, we need to determine the limits:

1. \(\lim_{x \to 2^-} f(x)\)
2. \(\lim_{x \to 2^+} f(x)\)

### Analyzing the Graph:
- The graph has two points of interest marked as \(P\) and \(Q\).
- As \(x\) approaches 2 from the left (\(2^-\)), the \(y\)-value of \(P\) appears to be the relevant limit. 
- As \(x\) approaches 2 from the right (\(2^+\)), the \(y\)-value of \(Q\) appears to be the relevant limit. 

### Observations:
- Point \(P\) has coordinates \((2, 4)\).
- Point \(Q\) has coordinates \((2, -2)\).

### Calculating the Limits:
1. **Left-hand limit** as \(x\) approaches 2:
   \[
   \lim_{x \to 2^-} f(x) = 4
   \]
2. **Right-hand limit** as \(x\) approaches 2:
   \[
   \lim_{x \to 2^+} f(x) = -2
   \]

These limits indicate a jump discontinuity at \(x = 2\), as the left-hand and right-hand limits are not equal.

Would you like to explore further details or have any other questions about this analysis?

**Related Questions:**
1. Can you describe what a jump discontinuity is?
2. Why is it important to check both left-hand and right-hand limits?
3. How would you evaluate a limit at a point where there is a hole in the graph?
4. What happens to the overall continuity of a function if there are jump discontinuities?
5. How would the function’s behavior change if the points \(P\) and \(Q\) were connected with a line?

**Tip:** When analyzing limits from a graph, it’s crucial to carefully observe where the function is approaching as \(x\) moves towards the point of interest from different directions.

Learn how to determine the left-hand and right-hand limits of a function from a graph as x approaches 2. This detailed explanation includes the analysis of limits, jump discontinuity, and graphical points. Perfect for students studying limits in Grade 11-12 or introductory calculus.

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