https://file.aiquickdraw.com/mathbot/file/1725942642283jmghy7lq.jpg

Let's analyze the graph and solve the limits and function values step by step. Based on the image you uploaded, I will interpret the graph and the questions as follows:

### Problem breakdown:
- (a) \(\lim_{x \to 2^-} f(x)\): The left-hand limit as \(x\) approaches 2 from the left.
- (b) \(\lim_{x \to 2^+} f(x)\): The right-hand limit as \(x\) approaches 2 from the right.
- (c) \(\lim_{x \to 2} f(x)\): The two-sided limit as \(x\) approaches 2.
- (d) \(f(2)\): The value of the function at \(x = 2\).
- (e) \(\lim_{x \to 4} f(x)\): The limit as \(x\) approaches 4.

### Analysis of the graph:

#### For \(x = 2\) (Parts a, b, c, d):
- From the graph, as \(x \to 2^-\), the function \(f(x)\) approaches \(0\).
- As \(x \to 2^+\), the function \(f(x)\) approaches \(2\).
- The two-sided limit \(\lim_{x \to 2} f(x)\) does not exist (DNE) because the left-hand and right-hand limits are not equal.
- \(f(2)\) appears to be \(f(2) = 0\), as the function touches the x-axis at \(x = 2\).

#### For \(x = 4\) (Part e):
- As \(x \to 4\), the function \(f(x)\) approaches \(2\). The graph is continuous around \(x = 4\), so the limit exists and is \(2\).

### Final Answers:
- (a) \(\lim_{x \to 2^-} f(x) = 0\)
- (b) \(\lim_{x \to 2^+} f(x) = 2\)
- (c) \(\lim_{x \to 2} f(x) = \text{DNE} \) (since the left and right limits are not equal)
- (d) \(f(2) = 0\)
- (e) \(\lim_{x \to 4} f(x) = 2\)

Would you like further clarification or more details on any part of this? 

Here are 5 related questions that can expand on the concepts discussed:
1. What conditions must be met for the two-sided limit \(\lim_{x \to c} f(x)\) to exist?
2. How do you find the limit of a function when given its graph?
3. What is the difference between a removable discontinuity and a jump discontinuity?
4. How can the value of a function at a point differ from its limit at that point?
5. Why does the limit not exist at \(x = 2\) in this case?

**Tip:** When dealing with piecewise functions or discontinuities, always check left-hand and right-hand limits separately to confirm the overall limit.

Use the given graph of f to state the value of each quantity, if it exists. (If an answer does not exist, enter DNE.)

(a) lim x → 2- f(x)
(b) lim x → 2+ f(x)
(c) lim x → 2 f(x)
(d) f(2)
(e) lim x → 4 f(x)

Learn how to evaluate limits and function values using a graph. This problem involves calculating left-hand and right-hand limits, determining if the overall limit exists, and evaluating the function value at specific points.

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