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Observe o gráfico abaixo que representa uma função 𝑓(𝑥) de ] − 3,4] em R, determine:

a) lim 𝑓(𝑥)
𝑥→1

b) lim 𝑓(𝑥)
𝑥→1−

c) lim 𝑓(𝑥) 
𝑥→1+


To solve these problems, we need to analyze the behavior of the function \( f(x) \) around the point \( x = 1 \) by looking at the graph you provided.

### a) \( \lim_{x \to 1} f(x) \)

This is the limit of \( f(x) \) as \( x \) approaches 1 from both sides (left and right). If the left-hand limit (as \( x \to 1^- \)) and the right-hand limit (as \( x \to 1^+ \)) are equal, then the two-sided limit exists and is equal to this value.

### b) \( \lim_{x \to 1^-} f(x) \)

This is the limit of \( f(x) \) as \( x \) approaches 1 from the left (values of \( x \) slightly less than 1). On the graph, we look at how \( f(x) \) behaves as \( x \) gets closer to 1 from the negative side.

### c) \( \lim_{x \to 1^+} f(x) \)

This is the limit of \( f(x) \) as \( x \) approaches 1 from the right (values of \( x \) slightly greater than 1). We analyze the graph to see the behavior of \( f(x) \) as \( x \) gets closer to 1 from the positive side.

---

**Analysis of the Graph:**

From the graph:

1. **Left-hand limit (as \( x \to 1^- \)):** As \( x \) approaches 1 from the left, the function \( f(x) \) is approaching the value \( y = -2 \). So:
   \[
   \lim_{x \to 1^-} f(x) = -2
   \]

2. **Right-hand limit (as \( x \to 1^+ \)):** As \( x \) approaches 1 from the right, the function \( f(x) \) is approaching the value \( y = 4 \). So:
   \[
   \lim_{x \to 1^+} f(x) = 4
   \]

3. **Two-sided limit (as \( x \to 1 \)):** Since the left-hand limit and right-hand limit are not equal, the two-sided limit does not exist:
   \[
   \lim_{x \to 1} f(x) \text{ does not exist.}
   \]

### Conclusion:

- **a) \( \lim_{x \to 1} f(x) \):** Does not exist.
- **b) \( \lim_{x \to 1^-} f(x) = -2 \)**
- **c) \( \lim_{x \to 1^+} f(x) = 4 \)**

Would you like to explore more details about these limits or have any specific questions? Here are a few related questions you might consider:

1. What is the significance of left-hand and right-hand limits in determining continuity?
2. How do we identify discontinuities in a function from its graph?
3. Can you describe how to find the limit of a function as \( x \to a \) when there are jumps in the graph?
4. How would the graph look if \( \lim_{x \to 1} f(x) \) existed?
5. What are different types of discontinuities and how do they affect limits?

**Tip:** When evaluating limits from a graph, always check for the behavior from both sides of the point in question.

Observe o gráfico abaixo que representa uma função 𝑓(𝑥) de ] − 3,4] em R, determine: a) lim 𝑓(𝑥) 𝑥→1 b) lim 𝑓(𝑥) 𝑥→1− c) lim 𝑓(𝑥) 𝑥→1+

Analyze the limits of a piecewise function as x approaches 1 from the left and right. The problem involves key concepts of limits and discontinuities, including left-hand and right-hand limits, with a step-by-step explanation.

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