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Let's analyze the function \( f(x) \) based on the graph and determine the required limits and function value at \( x = 2 \).

1. **Left-hand limit as \( x \to 2^- \):**
   The left-hand limit represents the value that \( f(x) \) approaches as \( x \) gets close to 2 from the left side. As we observe the graph for \( x < 2 \), the function is moving downward toward negative infinity as \( x \) approaches 2. 
   \[
   \lim_{x \to 2^-} f(x) = -\infty
   \]

2. **Right-hand limit as \( x \to 2^+ \):**
   The right-hand limit represents the value that \( f(x) \) approaches as \( x \) gets close to 2 from the right side. As we observe the graph for \( x > 2 \), the function is moving upward toward positive infinity as \( x \) approaches 2.
   \[
   \lim_{x \to 2^+} f(x) = +\infty
   \]

3. **Overall limit as \( x \to 2 \):**
   Since the left-hand limit is \( -\infty \) and the right-hand limit is \( +\infty \), the overall limit does not exist (DNE) at \( x = 2 \).
   \[
   \lim_{x \to 2} f(x) = \text{DNE}
   \]

4. **Value of the function at \( x = 2 \):**
   From the graph, there is no defined value for \( f(2) \), as there is a vertical asymptote at \( x = 2 \), meaning the function is not defined at this point.
   \[
   f(2) = \text{undefined}
   \]

Thus, the answers to the questions are:

- \( \lim_{x \to 2^-} f(x) = -\infty \)
- \( \lim_{x \to 2^+} f(x) = +\infty \)
- \( \lim_{x \to 2} f(x) = \text{DNE} \)
- \( f(2) = \text{undefined} \)

Would you like further details on any of these answers or explanations? Here are 5 related questions to consider:

1. What is the behavior of the function at other discontinuities in the graph?
2. How do limits relate to vertical asymptotes?
3. What is the difference between a limit existing and the function being undefined at a point?
4. How can we visually identify if a function has a jump discontinuity from a graph?
5. How do the left-hand and right-hand limits help in understanding piecewise functions?

**Tip:** Always check both the left-hand and right-hand limits when examining a limit at a point of discontinuity.

Answer all of the questions below about the function f(x) graphed below when x = 2.

Analyze the behavior of the function f(x) near the vertical asymptote at x = 2. This problem covers left-hand and right-hand limits, overall limits, and the value of the function. Suitable for high school calculus students.

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