https://file.aiquickdraw.com/mathbot/file/1725915600000juz9ldy9.jpg

Let's analyze the graph and answer the questions step by step based on the limits and function values.

### (a) \( \lim_{x \to 1} f(x) \)
- From the graph, as \( x \) approaches 1 from both sides, the function value approaches 2. Thus:
  \[
  \lim_{x \to 1} f(x) = 2
  \]

### (b) \( \lim_{x \to 3^-} f(x) \)
- As \( x \) approaches 3 from the left, the function value approaches 1. Thus:
  \[
  \lim_{x \to 3^-} f(x) = 1
  \]

### (c) \( \lim_{x \to 3^+} f(x) \)
- From the graph, as \( x \) approaches 3 from the right, the function seems to approach a value different from 1. It appears to approach 3. Thus:
  \[
  \lim_{x \to 3^+} f(x) = 3
  \]
  So, the correct answer for part (c) should be 3, not what was originally selected.

### (d) \( \lim_{x \to 3} f(x) \)
- Since the left-hand limit \( \lim_{x \to 3^-} f(x) = 1 \) and the right-hand limit \( \lim_{x \to 3^+} f(x) = 3 \), the limit does not exist because the left and right limits are not equal. Therefore:
  \[
  \lim_{x \to 3} f(x) = \text{DNE (Does Not Exist)}
  \]

### (e) \( f(3) \)
- From the graph, at \( x = 3 \), there is a filled-in point at \( y = 4 \), indicating that \( f(3) = 4 \). Therefore:
  \[
  f(3) = 4
  \]

The only corrections needed are for parts (c) and (e):
- \( \lim_{x \to 3^+} f(x) = 3 \)
- \( f(3) = 4 \)

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Do you want further clarification or details? Here are some related questions to deepen your understanding:

1. What is the difference between left-hand and right-hand limits?
2. Why does the limit not exist at \( x = 3 \)?
3. How do we determine the exact function value \( f(x) \) at a given point?
4. What are removable discontinuities, and does this graph contain any?
5. How can you verify limits algebraically if you are given a function formula instead of a graph?

**Tip:** When dealing with limits from graphs, always check whether the left-hand limit equals the right-hand limit to determine if the overall limit exists at a point.

For the function f whose graph is given, state the value of each quantity: limit as x approaches 1, limit from the left and right as x approaches 3, and the function value at x = 3.

Analyze the limit and continuity of a given function using graph interpretation. The question focuses on the limit as x approaches 1, the behavior of the function near x = 3 from both sides, and determining f(3). Suitable for high school calculus students.

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