https://file.aiquickdraw.com/mathbot/file/17265443175492mj7x2mz.jpg

Let's break down the graph and solve each part of the question step by step.

### Observing the Graph at \( x = 1 \):

- The function seems to have a **discontinuity** at \( x = 1 \).
- There's a **jump** discontinuity, which means the left-hand limit (\( \lim_{x \to 1^-} f(x) \)) and the right-hand limit (\( \lim_{x \to 1^+} f(x) \)) are different.

Now, let's compute each value:

### 1. \( \lim_{x \to 1^-} f(x) \):
This is the limit as \( x \) approaches 1 from the left. Looking at the graph, as \( x \) approaches 1 from the left side (from values like \( x = 0.5, 0.9 \)), the function tends to a value of **3**. 

Thus, 
\[
\lim_{x \to 1^-} f(x) = 3
\]

### 2. \( \lim_{x \to 1^+} f(x) \):
This is the limit as \( x \) approaches 1 from the right. From the graph, as \( x \) approaches 1 from the right side (from values like \( x = 1.1, 1.5 \)), the function tends to a value of **-4**.

Thus, 
\[
\lim_{x \to 1^+} f(x) = -4
\]

### 3. \( \lim_{x \to 1} f(x) \):
The overall limit at \( x = 1 \) exists only if both the left-hand and right-hand limits are equal. Since \( \lim_{x \to 1^-} f(x) = 3 \) and \( \lim_{x \to 1^+} f(x) = -4 \), these are **not equal**, so the limit **does not exist**.

Thus, 
\[
\lim_{x \to 1} f(x) = \text{DNE (Does Not Exist)}
\]

### 4. \( f(1) \):
The value of the function at \( x = 1 \) is given by the point at \( x = 1 \). From the graph, there is a **solid dot** at \( x = 1 \) corresponding to the value **-2**.

Thus, 
\[
f(1) = -2
\]

### Summary of Answers:
1. \( \lim_{x \to 1^-} f(x) = 3 \)
2. \( \lim_{x \to 1^+} f(x) = -4 \)
3. \( \lim_{x \to 1} f(x) = \text{DNE} \)
4. \( f(1) = -2 \)

Let me know if you want more details or have further questions!

Here are 5 related questions:
1. What is the difference between jump and removable discontinuities?
2. How would the graph change if \( \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) \)?
3. Can a function have a value at \( x = 1 \) even if \( \lim_{x \to 1} f(x) \) does not exist?
4. How do you calculate the limit at a point of continuity versus discontinuity?
5. How do you graphically identify the left-hand and right-hand limits?

**Tip**: When analyzing a discontinuous function, always check the left-hand and right-hand limits separately to identify the type of discontinuity!

Solve for the limits and the value of the function f(x) when x = 1, based on the graph provided.

Understand how to solve for limits and identify discontinuities at x = 1 using a graphical representation. This problem covers key concepts such as left-hand and right-hand limits, jump discontinuities, and how to compute function values.

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