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Let's go through this problem systematically:

The image shows a graph of a function \( f(x) \) and a table asking for several limits at specific points. It also includes some questions about continuity and types of discontinuities.

### Step 1: Addressing the Limits Table

Based on the graph:
- **\(\lim_{x \to -2^-} f(x)\):** As \(x\) approaches \(-2\) from the left, the function seems to approach negative infinity. Therefore, \(\lim_{x \to -2^-} f(x) = -\infty\).
- **\(\lim_{x \to -2^+} f(x)\):** As \(x\) approaches \(-2\) from the right, the function approaches positive infinity. So, \(\lim_{x \to -2^+} f(x) = +\infty\).
- **\(\lim_{x \to -2} f(x)\):** Since the limits from both sides differ (one goes to \(-\infty\), and the other goes to \(+\infty\)), the two-sided limit does not exist (DNE).

- **\(\lim_{x \to -1^-} f(x)\):** As \(x\) approaches \(-1\) from the left, it appears to approach \(0\). So, \(\lim_{x \to -1^-} f(x) = 0\).
- **\(\lim_{x \to -1^+} f(x)\):** As \(x\) approaches \(-1\) from the right, the function also approaches \(0\). Hence, \(\lim_{x \to -1^+} f(x) = 0\).
- **\(\lim_{x \to -1} f(x)\):** Since both one-sided limits equal \(0\), the two-sided limit exists and equals \(0\).

- **\(\lim_{x \to 0^-} f(x)\):** As \(x\) approaches \(0\) from the left, the function approaches \(1\). So, \(\lim_{x \to 0^-} f(x) = 1\).
- **\(\lim_{x \to 0^+} f(x)\):** As \(x\) approaches \(0\) from the right, the function approaches \(-1\). So, \(\lim_{x \to 0^+} f(x) = -1\).
- **\(\lim_{x \to 0} f(x)\):** Since the one-sided limits differ, the two-sided limit does not exist (DNE).

- **\(\lim_{x \to 3^-} f(x)\):** As \(x\) approaches \(3\) from the left, the function approaches \(2\). So, \(\lim_{x \to 3^-} f(x) = 2\).
- **\(\lim_{x \to 3^+} f(x)\):** As \(x\) approaches \(3\) from the right, the function also approaches \(2\). Hence, \(\lim_{x \to 3^+} f(x) = 2\).
- **\(\lim_{x \to 3} f(x)\):** Since both one-sided limits equal \(2\), the two-sided limit exists and equals \(2\).
- **\(f(3)\):** From the graph, \(f(3)\) seems to be \(1\), since there is a hole at \(x=3\) with a defined point below the hole. So, \(f(3) = 1\).

### Step 2: Vertical Asymptotes
A vertical asymptote occurs where the function approaches infinity or negative infinity as \(x\) approaches a specific value. Based on the graph, \( f(x) \) has a vertical asymptote at:
- \(x = -2\), since the function goes to \(+\infty\) and \(-\infty\) around \(x = -2\).

### Step 3: Continuity
The function is continuous where there are no jumps, holes, or vertical asymptotes.
- **Intervals of continuity:** Based on the graph, \( f(x) \) is continuous on the intervals \((- \infty, -2)\), \((-1, 0)\), \((0, 3)\), and \((3, \infty)\). 

### Step 4: Discontinuities
Dis

Consider the graph of the function f(x) below and answer the following questions. If you are asked to find a value that does not exist (and is not infinite), write DNE. Include equal signs where appropriate. State all intervals on which f(x) is continuous and categorize any discontinuities.

This problem covers how to evaluate limits, determine points of continuity, and identify discontinuities using the graph of a function. Learn how to apply limit laws, recognize vertical asymptotes, and classify discontinuities as jump, infinite, or removable.

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