https://file.aiquickdraw.com/mathbot/file/17270551523845v85isb4.jpg

Let's analyze the graph of the function \( f(x) \) and evaluate the limits based on the information given in the image.

### Step-by-step Analysis

#### 1. \(\lim_{x \to 3^-} f(x)\)
As \( x \) approaches 3 from the left:
- The value of \( f(x) \) is approaching 2.

#### 2. \(\lim_{x \to 3^+} f(x)\)
As \( x \) approaches 3 from the right:
- The value of \( f(x) \) is approaching 3.

Since the left-hand and right-hand limits are not equal, the overall limit at \( x = 3 \) does not exist.

#### 3. \(\lim_{x \to -1^-} f(x)\)
As \( x \) approaches \(-1\) from the left:
- The value of \( f(x) \) is approaching 0.

#### 4. \(\lim_{x \to -1^+} f(x)\)
As \( x \) approaches \(-1\) from the right:
- The value of \( f(x) \) is approaching 1.

Since the left-hand and right-hand limits are not equal, the overall limit at \( x = -1 \) does not exist.

#### 5. \(\lim_{x \to 4} f(x)\)
As \( x \to 4 \), there is a jump in the graph:
- From the left, \( f(x) \to 3 \).
- From the right, \( f(x) \to 1 \).

Since the left-hand and right-hand limits do not match, the limit does not exist.

### Answer Summary
\[
\begin{aligned}
&\lim_{x \to 3^-} f(x) = 2, \\
&\lim_{x \to 3^+} f(x) = 3, \\
&\lim_{x \to 3} f(x) = \text{DNE}, \\
&\lim_{x \to -1^-} f(x) = 0, \\
&\lim_{x \to -1^+} f(x) = 1, \\
&\lim_{x \to -1} f(x) = \text{DNE}, \\
&\lim_{x \to 4} f(x) = \text{DNE}.
\end{aligned}
\]

Would you like me to provide further details or answer any specific questions?

### 5 Related Questions
1. What is a one-sided limit and how does it differ from a two-sided limit?
2. How do you determine if a limit exists at a given point?
3. What happens when the left-hand and right-hand limits do not match at a point?
4. How can discontinuities like jumps affect the existence of limits?
5. Why is it important to analyze limits from both sides for piecewise or step functions?

### Tip
Always check both left-hand and right-hand limits when dealing with graphs, as this helps identify points of discontinuity where a limit may not exist.

Determine the values of limits from a graph of f(x), focusing on x = 3, x = -1, and x = 4.

Learn how to evaluate one-sided and two-sided limits from a graph with discontinuities. This problem involves determining the limits as x approaches specific points (x=3, x=-1, and x=4), where the graph has jump discontinuities. Suitable for high school calculus or college-level students.

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