https://file.aiquickdraw.com/mathbot/file/1727398480508a5ddv9i3.jpg

Let's analyze the problem depicted in the image. We have a piecewise linear function \( f(x) \) represented by its graph, and the task is to evaluate the following limits:

1. **(a) \( \lim_{x \to 2} f(x) \)**
   - This is the overall limit of the function as \( x \) approaches 2.

2. **(b) \( \lim_{x \to 2^+} f(x) \)**
   - This is the right-hand limit of the function as \( x \) approaches 2 from the right.

3. **(c) \( \lim_{x \to 2^-} f(x) \)**
   - This is the left-hand limit of the function as \( x \) approaches 2 from the left.

### Step 1: Analyze the graph around \( x = 2 \)

Looking at the graph:
- For \( x < 2 \), the graph is a straight line with a negative slope.
- For \( x > 2 \), the graph is a straight line with a positive slope.

#### Left-hand limit: \( \lim_{x \to 2^-} f(x) \)
From the left (as \( x \) approaches 2), the function appears to approach the value \( y = 1 \). Hence:
\[
\lim_{x \to 2^-} f(x) = 1
\]

#### Right-hand limit: \( \lim_{x \to 2^+} f(x) \)
From the right (as \( x \) approaches 2), the function also appears to approach the value \( y = 1 \). Hence:
\[
\lim_{x \to 2^+} f(x) = 1
\]

#### Overall limit: \( \lim_{x \to 2} f(x) \)
Since both the left-hand and right-hand limits are equal, the overall limit exists and is also 1. Therefore:
\[
\lim_{x \to 2} f(x) = 1
\]

### Final Answers:
- **(a) \( \lim_{x \to 2} f(x) = 1 \)**
- **(b) \( \lim_{x \to 2^+} f(x) = 1 \)**
- **(c) \( \lim_{x \to 2^-} f(x) = 1 \)**

Would you like more detailed explanations, or do you have any specific questions? 

Here are 5 related questions to explore further:
1. What is the formal definition of a limit?
2. How do we handle limits when the left-hand and right-hand limits differ?
3. What is a piecewise function, and how does it affect limit calculations?
4. How can you determine continuity from a graph like this?
5. How are limits related to derivatives?

**Tip:** When evaluating limits from a graph, always check both the left-hand and right-hand limits for any discontinuities.

For the piecewise linear function f(x) graphed below, find the following limits:
(a) lim x->2 f(x)
(b) lim x->2^+ f(x)
(c) lim x->2^- f(x)

Solve for the limits of a piecewise linear function as x approaches 2. This problem involves analyzing the left-hand and right-hand limits from a graph. Suitable for high school students learning about limits and piecewise functions.

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