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Dadas las siguientes gráficas de funciones, determine los límites laterales en el valor de 
X indicado y analice la existencia del límite. 

Let's analyze the given graph to determine the lateral limits and assess the existence of the limit at \( x = 4 \).

### 1. Identifying the Function Behavior:
The graph shows a piecewise linear function that is defined over two intervals:
- For \( x < 4 \), the graph is increasing.
- For \( x > 4 \), the graph is decreasing.

There is a point marked on the graph at \( (4, 0) \) with the label "En \( y = 4 \)," indicating that the function has a jump at this point.

### 2. Left-Hand Limit (\( \lim_{x \to 4^-} f(x) \)):
- As \( x \) approaches 4 from the left (\( x \to 4^- \)), the function value rises up to the peak at \( y = 3 \).
- Therefore, the left-hand limit is:
  \[
  \lim_{x \to 4^-} f(x) = 3
  \]

### 3. Right-Hand Limit (\( \lim_{x \to 4^+} f(x) \)):
- As \( x \) approaches 4 from the right (\( x \to 4^+ \)), the function value descends from the peak back down to 2.
- Therefore, the right-hand limit is:
  \[
  \lim_{x \to 4^+} f(x) = 2
  \]

### 4. Assessing the Existence of the Limit at \( x = 4 \):
- The limit exists at \( x = 4 \) if and only if the left-hand limit equals the right-hand limit.
- Since \( \lim_{x \to 4^-} f(x) = 3 \) and \( \lim_{x \to 4^+} f(x) = 2 \), the left and right limits are not equal.
- Therefore, the overall limit at \( x = 4 \) does not exist.

### Conclusion:
- **Left-Hand Limit:** \( \lim_{x \to 4^-} f(x) = 3 \)
- **Right-Hand Limit:** \( \lim_{x \to 4^+} f(x) = 2 \)
- **Overall Limit:** Does not exist at \( x = 4 \).

Would you like more details, or do you have any questions?

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**Relative Questions:**
1. What is the significance of lateral limits in analyzing the behavior of functions at specific points?
2. How would you describe the difference between a jump discontinuity and a removable discontinuity?
3. What are the implications of a non-existent limit in real-world applications?
4. How can the concept of lateral limits be extended to functions with more complex behaviors, such as oscillations?
5. Can you think of an example where the left-hand limit and right-hand limit are equal but the overall limit does not exist?

**Tip:** When dealing with piecewise functions, always pay attention to the behavior of the function on each side of the point of interest to correctly assess the existence of the limit.

Explore how to determine lateral limits and assess the existence of limits in functions using a detailed graphical analysis. This problem involves piecewise linear functions and examines the behavior of functions around specific points. Suitable for advanced high school students.

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